Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.
Answers
Answer:
[tex]\dfrac{4096\pi}{5}\approx 2573.593\; \sf (3\;d.p.)[/tex]
Step-by-step explanation:
The shell method is a calculus technique used to find the volume of a solid revolution by decomposing the solid into cylindrical shells. The volume of each cylindrical shell is the product of the surface area of the cylinder and the thickness of the cylindrical wall. The total volume of the solid is found by integrating the volumes of all the shells over a certain interval.
The volume of the solid formed by revolving a region, R, around a vertical axis, bounded by x = a and x = b, is given by:
[tex]\displaystyle 2\pi \int^b_ar(x)h(x)\;\text{d}x[/tex]
where:
r(x) is the distance from the axis of rotation to x.h(x) is the height of the solid at x (the height of the shell).[tex]\hrulefill[/tex]
We want to find the volume of the solid formed by rotating the region bounded by y = 0, y = √x, x = 0 and x = 16 about the y-axis.
As the axis of rotation is the y-axis, r(x) = x.
Therefore, in this case:
[tex]r(x)=x[/tex]
[tex]h(x)=\sqrt{x}[/tex]
[tex]a=0[/tex]
[tex]b=16[/tex]
Set up the integral:
[tex]\displaystyle 2\pi \int^{16}_0x\sqrt{x}\;\text{d}x[/tex]
Rewrite the square root of x as x to the power of 1/2:
[tex]\displaystyle 2\pi \int^{16}_0x \cdot x^{\frac{1}{2}}\;\text{d}x[/tex]
[tex]\textsf{Apply the exponent rule:} \quad a^b \cdot a^c=a^{b+c}[/tex]
[tex]\displaystyle 2\pi \int^{16}_0x^{\frac{3}{2}}\;\text{d}x[/tex]
Integrate using the power rule (increase the power by 1, then divide by the new power):
[tex]\begin{aligned}\displaystyle 2\pi \int^{16}_0x^{\frac{3}{2}}\;\text{d}x&=2\pi \left[\dfrac{2}{5}x^{\frac{5}{2}}\right]^{16}_0\\\\&=2\pi \left[\dfrac{2}{5}(16)^{\frac{5}{2}}-\dfrac{2}{5}(0)^{\frac{5}{2}}\right]\\\\&=2 \pi \cdot \dfrac{2}{5}(16)^{\frac{5}{2}}\\\\&=\dfrac{4\pi}{5}\cdot 1024\\\\&=\dfrac{4096\pi}{5}\\\\&\approx 2573.593\; \sf (3\;d.p.)\end{aligned}[/tex]
Therefore, the volume of the solid is exactly 4096π/5 or approximately 2573.593 (3 d.p.).
[tex]\hrulefill[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\underline{Power Rule of Integration}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}(+\;\text{C})$\\\end{minipage}}[/tex]
Related Questions
Which expressions are equivalent to 8(3/4y - 2) + 6(-1/2x + 4) + 1
Answers
Answer:
8(3/4y - 2) + 6(-1/2x + 4) + 1 can be simplified as:
6(-1/2x) = -3x
8(3/4y) = 6y
8(-2) = -16
6(4) = 24
1 remains as 1.
So the expression becomes:
6y - 3x - 16 + 24 + 1
which simplifies to:
6y - 3x + 9
Therefore, the expressions that are equivalent to 8(3/4y - 2) + 6(-1/2x + 4) + 1 are:
6y - 3x + 9
Hydrogen produced from a hydrolysis reaction was collected over water. The data is compiled in the table.
Total volume of H2(g) collected 94.00 mL
Temperature 26.0 °C
Barometric pressure 745 mmHg
Vapor pressure of water at 26.0 °
25.5 mmHg
Calculate the moles of hydrogen gas produced by the reaction.
moles:
Answers
Answer:
To calculate the moles of hydrogen gas produced, we need to use the ideal gas law equation:
PV = nRT
where P is the total pressure, V is the volume of the gas, n is the number of moles of gas, R is the gas constant, and T is the temperature in Kelvin.
First, we need to convert the temperature to Kelvin by adding 273.15:
T = 26.0 + 273.15 = 299.15 K
Next, we need to calculate the pressure of the hydrogen gas by subtracting the vapor pressure of water from the barometric pressure:
P(H2) = P(total) - P(vapor) = 745 - 25.5 = 719.5 mmHg
Now we can plug in the values into the ideal gas law equation and solve for n:
n = PV/RT
n = (719.5 mmHg)(94.00 mL)/(62.3637 L mmHg/K mol)(299.15 K)
n = 0.00384 mol
Therefore, the moles of hydrogen gas produced by the reaction is 0.00384 mol.
HELP PLEASEEE 30 POINTS!!
Answers
Answer:
m<1 = 63° (Exterior alternating Angles)
m<2 = 62°
m<3 = 118°
Step-by-step explanation:
[tex]{ \tt{m \angle 2 + 63 \degree + 55 \degree = 180 \degree}} \\ { \sf{(exterior \: corresponding \: angles)}} \\ { \tt{m \angle 2 = 180 - (63 + 55)}} \\ { \tt{ \underline{ \: m \angle 2 = 62 \degree \: }}}[/tex]
[tex]{ \tt{m \angle 3 = m \angle 1 + 55 \degree}} \\ { \tt{m \angle 3 = 63 + 55}} \\ { \tt{ \underline{ \: m \angle 3 = 118 \degree \: }}}[/tex]
HELP! WILL MARK BRAINLIEST!!! Inverse Functions Digital Equations
Answers
Answer:
u8hiuhihihoiho'hjoihuiytrdeee
Step-by-step explanation:
I'm not sure where I'm going wrong here and need a detailed explanation with a full answer please.
Answers
Answer:
g(x) = 1/2|x - 1| + 2
Step-by-step explanation:
You were almost there, you just got the slope wrong.
original vertex (h, k): (0, 0)
transformed vertex (h', k'): (1, 2)
original slope: 1
transformed slope: m = (3-2)/(3-1) = 1/2 pick any 2 points to find the slope
equation for the transformed function shown in the graph:
g(x) = 1/2|x - h'| + k'
g(x) = 1/2|x - 1| + 2
Between 11pm and midnight on Thursday night Mystery pizza gets an average of 4.2 telephone orders per hour
A. Find the probability that at least 3 minutes will elapse before the next telephone order
B. Find the probability that less then 15 minutes will elapse
C. Find the probability that between 15 and 30 minutes will elapse
Answer all please URGENT
Answers
The probability that at least 3 minutes will elapse before the next telephone order is 0.797.
The probability that less than 15 minutes will elapse between orders is 0.677.
The probability that between 15 and 30 minutes will elapse between orders is 0.2275
Using Poisson distribution:To solve the following problem, we need to use the Poisson distribution, which is a probability distribution that describes the number of events that occur in a fixed interval of time or space, given the average rate of occurrence of those events.
The Poisson distribution has the following formula:
[tex]P(X = k) = (\lambda\times ex^{-\lambda}) / k![/tex]
Where:
P(X = k) is the probability that there are exactly k events in the interval
λ is the average rate of occurrence of events in the interval
e is the mathematical constant e (approximately 2.71828)
k! is the factorial of k (i.e., k * (k-1) * (k-2) * ... * 2 * 1)
Here we have
Between 11 pm and midnight on Thursday night Mystery pizza gets an average of 4.2 telephone orders per hour
A. The probability that at least 3 minutes will elapse before the next telephone order, using the complement rule:
=> P(at least 3 minutes) = 1 - P(less than 3 minutes)
Assume that the time between telephone orders follows an exponential distribution with a mean of 1/4.2 = 0.2381 hours (or 14.28 minutes).
Therefore, the Poisson distribution is λ = 1/0.2381 = 4.2/1.0 = 4.2.
Using the exponential distribution, we can find the probability of less than 3 minutes elapsing between orders as follows:
P(less than 3 minutes) = [tex]1 - e ^{(-\lambda \times t) }[/tex]
Where t = 3/60 = 0.05 hours
P(less than 3 minutes) = [tex]1 - e^{(-4.2\times 0.05) } = 0.203[/tex]
Therefore,
P(at least 3 minutes) = 1 - 0.203 = 0.797
The probability that at least 3 minutes will elapse before the next telephone order is 0.797.
B. To find the probability that less than 15 minutes will elapse between orders, we can use the same exponential distribution as before and set t = 15/60 = 0.25 hours:
P(less than 15 minutes) = [tex]1 - e ^{(-\lambda \times t) }[/tex]
P(less than 15 minutes) = [tex]1 - e^{(-4.2 \times 0.25)} = 0.677[/tex]
Hence, The probability that less than 15 minutes will elapse between orders is 0.677.
C. To find the probability that between 15 and 30 minutes will elapse between orders, we can subtract the probabilities found in less than 15 minutes and less than 30 minutes.
P(15 to 30 minutes) = P(less than 15 minutes) - P(less than 30 minutes) -
P(15 to 30 minutes) = [tex]e^{ (-\lambda0.5)} - e^{ (-\lambda 0.25)}[/tex]
= 0.3499 - 0.1224 = 0.2275
Therefore,
The probability that at least 3 minutes will elapse before the next telephone order is 0.797.
The probability that less than 15 minutes will elapse between orders is 0.677.
The probability that between 15 and 30 minutes will elapse between orders is 0.2275
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Multiply: (–4 + i)(2 – 3i)
Answers
Answer:
Step-by-step explanation:
We can use the distributive property and FOIL method to multiply these two complex numbers:
(-4 + i)(2 - 3i) = -4(2) + (-4)(-3i) + (i)(2) + (i)(-3i)
Simplifying each term, we get:
(-4 + i)(2 - 3i) = -8 + 12i + 2i - 3i^2
Since i^2 = -1, we can replace it with -1:
(-4 + i)(2 - 3i) = -8 + 12i + 2i - 3(-1)
Simplifying further, we get:
(-4 + i)(2 - 3i) = -5 + 14i
Therefore, the product of (-4 + i) and (2 - 3i) is -5 + 14i.
Sanborn Inc. is a new manufacturing company founded on February 2, 2012. The company had to choose between the LIFO and FIFO methods for its inventory. Inventory costs were rising during 2012, so the company decided to use the LIFO method. Which of the following items would be decreased by the choice of LIFO (compared to what would have happened if they chose to use FIFO)? (check all that apply)
Answers
Answer:
Step-by-step explanation:
By choosing the LIFO method for inventory, the following items would be decreased (compared to what would have happened if they chose to use FIFO):
Net income
Taxes payable
Total assets
Owner's equity
This is because the LIFO method results in higher cost of goods sold and lower ending inventory, which in turn decreases net income and taxes payable. Lower ending inventory also decreases total assets and owner's equity.
The items that would be decreased by the choice of LIFO (compared to what would have happened if they chose to use FIFO) are Ending inventory and Net profit
What is the unitary method?
The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value. Unitary method is a technique by which we find the value of a single unit from the value of multiple devices and the value of more than one unit from the value of a single unit. It is a method that we use for most of the calculations in math.
We are given that;
Inventory costs in 2012 were rising
Now,
LIFO stands for last-in, first-out, and FIFO stands for first-in, first-out. These are two ways of assigning costs to the units of inventory that you sell. FIFO assumes that you sell the oldest units first, while LIFO assumes that you sell the newest units first1.
The choice of LIFO or FIFO has implications on a company’s financial statements as this decision impacts the value of inventory, cost of goods sold, and net profit2. If inventory costs are rising during a period, then LIFO will result in a lower value of ending inventory, a higher cost of goods sold, and a lower net profit compared to FIFO1.
Therefore, by unitary method the answer will be Ending inventory and Net profit
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the lotka-volterra prey-predator equations with self-limitation ofprey are given in non-dimensional form by dt
Answers
The parameter an in the Lotka-Volterra predator-prey model denotes the capture efficiency (alpha).
The lotka-Volterra predator-prey model is what.
The Lotka-Volterra model makes the assumption that a predator's rate of prey consumption is inversely related to its abundance.
The parameter an in the Lotka-Volterra predator-prey model denotes the capture efficiency (alpha).
In conclusion, the lotka-volterra predator-prey model aids in estimating the rate at which prey is consumed.
We possess that
The predator population is x.
y is the number of prey.
The dy/dt equation contains the coefficient c. Hence, this coefficient is correlated with the prey population. The presence of a minus sign indicates a decline in the population of prey. It multiplies x as well, indicating a connection to the predator population.
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Answer the question attached above. The first person who gets it right will be given brainly.
Answers
Answer:
Step-by-step explanation:
a) One possible function that satisfies the conditions is:
f(x) = (x^2 - 25) / (x - 5)
This function is undefined when x = 5, but it is defined for all other real numbers.
b) One possible function that satisfies the conditions is:
g(x) = 1 / (x - 1)
This function is undefined when x = 1, but it is defined for all other positive numbers greater than 1.
c) One possible function that satisfies the conditions is:
h(x) = ln(x - 1)
This function is undefined when x = 1, but it is defined for all other positive numbers greater than 1. The range of this function is all real numbers.
Which quadratic function best fits the data?
x y
1 120
2 75
3 47
4 35
5 47
6 75
Answers
The quadratic function that best matches the provided data is as follows: y = -9.38x2 - 74.8x + 186.1 with a 1594.61 squared difference total.
what is function ?A function in mathematics is a connection between two groups of numbers, known as the domain and the range, where each element in the domain has a unique relationship with each element in the range. The range is the set of possible output values that the function can generate, and the domain is the set of input values for which the function is defined. Algebraic expressions or formulas that explain how the output values are arrived at based on the input values are frequently used to depict functions.
given
The greatest fit will be determined by the function with the smallest sum of squared differences.
The outcome is as follows after computing the sums of squared differences for each function:
Sum of squared differences for y = 9.38x2 + 74.8x + 186.1 is 406.27.
Sum of squared differences for y = -9.38x2 + 74.8x + 186.1 is 406.27.
Sum of squared differences for y = 9.38x2 + 274.8x + 186.1 is 18681.95.
sum of squared disparities is 1594.61 when y = -9.38x2 - 74.8x + 186.1.
The quadratic function that best matches the provided data is as follows: y = -9.38x2 - 74.8x + 186.1 with a 1594.61 squared difference total.
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The breadth of a rectangular playground is 5m shorter than its length. If its perimeter is 130m,find ids length and breadth.
Answers
Answer:
Length is 35 m and breadth is 30 mStep-by-step explanation:
Given,
The breadth of a rectangular playground is 5m shorter than its length.Perimeter is 130 mLet length be x and breadth (x - 5).
Perimeter of rectangle is calculated by :
[tex] \: \: \boxed{ \pmb{ \sf{Perimeter_{(rectangle)} = 2(l + b)}}} \\ [/tex]
On substituting the values we get :
[tex]\dashrightarrow \: \: 130 = 2(x + x - 5) \\ [/tex]
[tex]\dashrightarrow \: \: 130 = 2(2x - 5) \\ [/tex]
[tex]\dashrightarrow \: \dfrac{130}{2} = (2x - 5) \\ [/tex]
[tex]\dashrightarrow \: \: 65 = 2x - 5 \\ [/tex]
[tex]\dashrightarrow \: \: 65 + 5 = 2x \\ [/tex]
[tex]\dashrightarrow \: \: 70 = 2x \\ [/tex]
[tex]\dashrightarrow \: \: \frac{70}{2} = x \\ [/tex]
[tex]\dashrightarrow \: \: 35 = x \\ [/tex]
Hence,
Length = x = 35 m.Breadth = x -5 = (35 -5) = 30 mPLS HELP FAST IM GIVING 50 POINTS
Answers
Answer:
4 + - 6 = -2
Hope this helps!
Step-by-step explanation:
The arrow closest to the line shows going forward 4 or + 4
The second arrow shows going back 6 or -6
+ 4 - 6 = -6 + 4 = -2
place the following steps in order. what is the correct order for the general procedure for hypothesis testing?
Answers
The correct order of the steps to test the general hypothesis is given as B, A, E, C, D, F.
A hypothesis is basically a uncertain fact that is backed by some logical and solid information. While using the hypothesis, it s important that we use the correct method to verify the credibility of the hypothesis in the system.
So, as per the given option, the correct sequence should be, B, A, E, C, D, F.
We first set up the null hypothesis and the alternative hypothesis, and then we choose the degree of significance. The next phases in hypothesis testing are choosing the test statistics and creating the decision-making rule.
When we get to a conclusion based on the hypothesis after carefully calculating and computing the hypothesis' performance, the procedure is said to be finished.
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Complete question - place the following steps in order. what is the correct order for the general procedure for hypothesis testing?
A. Select the level of significance
B. setup null and alternative hypothesis
C. Establish the decision rule
D. performance computation
E. select the test statistics
F. Draw conclusions
A spinner with 10 equally sized slices has 5 red slices, 3 yellow slices, and 2 blue slices. Ann spun the dial 25 times. It landed on red 12 times, landed on yellow 10 times, and landed on blue 3 times. From Ann's results, compute the experimental probability of landing on blue or yellow
Answers
Answer:
0.600 or 600, 0.500 or 500, select option 2
Step-by-step explanation:
Question 2 of 3
Which subtraction equation shows how to subtract
4
2
12
−
2
8
12
using equivalent fractions? i need help
Answers
Answer:
Step-by-step explanation:
your given is not cleared repost it then post
f(x)
g(x)
=6x−4
=3x
2
−2x−10
Escribe (g∘f)(x) como una expresión en términos de x.
Answers
Answer:
g(x)
=6x−4
=3x
2
−2x−10
Escribe (g∘f)(x) como una expresión en términos de x.
Step-by-step explanation:
Primero necesitamos conocer la función f(x). Luego podemos sustituir f(x) en g(x) para obtener (g∘f)(x).
Como la función f(x) no se proporcionó en la pregunta, asumiré que f(x) es:
f(x) = x^2 - 2x + 1
Entonces, podemos sustituir f(x) en g(x) de la siguiente manera:
g(f(x)) = 6f(x) - 4
= 6(x^2 - 2x + 1) - 4 (sustituyendo f(x))
= 6x^2 - 12x + 2
Por lo tanto, (g∘f)(x) = 6x^2 - 12x + 2.
Highlight (1)
Science - Ruhul A Mollah
W-
OD.
South
-E Pacific
Ocean
South
Atlantic
Ocean
38ee7c7-4516-4260-a430-9d81b38467df
Search
Indian
Ocean
3
Southern
Ocean
LEGEND
= increasing sea level trends
•= decreasing sea level trends
Do the local tide measurement data support or contradict the global satellite data?
OA.
The tide measurement data contradict the satellite data because the tide measurement data show that the local sea level trends at some locations
are decreasing.
OB. The tide measurement data support the satellite data because the number of places where local sea level trends are decreasing is so small that they
can be ignored.
OC. The tide measurement data contradict the satellite data because the number of decreasing local sea level trends is greater than the number of
increasing local sea level trends.
8
(3
S
The tide measurement data support the satellite data because the tide measurement data show that the number of increasing local sea level trends a
significantly greater than the number of decreasing local sea level trends.
ex
#
Time Remaining 00:30:14
Question 10 of 16
A Bag for Review
Previous
AOGRE0
Answers
The local tide measurement data contradict the satellite data because the number of decreasing local sea level trends is greater than the number of increasing local sea level trends. So correct option is C.
Describe Measuring of tides?Measuring tides involves observing and recording changes in the water level of oceans, seas, and other bodies of water over time. Tides are caused by the gravitational pull of the moon and sun on the Earth's water, which results in a rhythmic rise and fall of water levels.
Tide measurements are important for a variety of reasons, including understanding ocean circulation, predicting flooding and storm surges, and planning shipping routes. Tidal data is also used in climate research, as changes in sea level can indicate long-term climate trends.
The local tide measurement data contradict the satellite data because the number of decreasing local sea level trends is greater than the number of increasing local sea level trends.
The measurement is done because the land surfaces are dynamic with some locations rising or sinking relative to sea level changes are different across world coasts.
Therefore, the Option C is correct.
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The complete question is:
In which three statements below will the number 8 correctly fill
in the blank?
Answers
the correct options are A), B), and E).
why it is and what is Gallon?
A) 2 quarts = 8 cups
B) 8 cm = 80mm
E) 96 inches = 8 feet
The number 8 cannot correctly fill in the blank for statements C and D.
C) 1 gallon = 16 cups, so 4 gallons = 64 cups, not 8 cups.
D) 1 hour = 60 minutes, so 96 minutes = 1 hour and 36 minutes, not 8 hours.
Therefore, the correct answers are A), B), and E).
A gallon is a unit of measurement for volume commonly used in the United States and some other countries. There are two different sizes of gallons: the US gallon and the imperial gallon.
The US gallon is defined as exactly 3.785411784 liters, and is used for measuring liquids such as gasoline, milk, and other beverages.
The imperial gallon, which is used in the United Kingdom and some other countries, is defined as exactly 4.54609 liters.
In both cases, a gallon is typically divided into smaller units such as quarts, pints, and fluid ounces for measuring smaller amounts of liquid.
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If someone can help with ALL the questions that would be greatly appreciated. The last answer I got for this post only answered the first question. Which is $60, but I really need help with all of them. This subject has been difficult for me to grasp. Thank you all for your help.
1. Avril deposited $800 in his bank at 1.5% for five years.
a) Use the simple interest formula to calculate the
amount of interest Avril will earn.
b) Calculate the future value of her deposit.
2. Phyllis borrowed $1,000 at 2% APR for six months. Use the simple interest formula to calculate the total amount of interest Phyllis will pay if she pays $200 two months into the loan and the rest at six months.
3. James deposited $500 in his bank at 4.5% for 90 days.
a) Use the simple interest formula to calculate the
amount of interest James will earn using exact
interest.
b) Calculate the future value of the deposit.
4. On May 4 Ariel signed a simple discount note for $3,500 at 3 1/2% for 60 days.
a) Use the simple interest formula to calculate
the amount of interest Ariel will pay using
ordinary interest.
b) Calculate the proceeds she will receive on
May 4.
c) Calculate the amount she will pay at maturity
d) Determine the date it will be due.
5. What is the APY (effective rate) for 16% APR compounded quarterly? Round to hundredths.
6. Louisa invested $4,500 at 4% interest compounded semiannually for two years. Calculate the future value of Louisa's investment using Exhibit 11-1 or the formula FV = P(1 + R)n.
7. Cleve wants to know how much he would need to deposit now in order to have $8,000 in five years at a rate of 6% compounding quarterly. Use Exhibit 11-2 or the formula P = FV/(1 + R)n.
8. I want to borrow $900 for 20 days from a payday loan store. The payday loan finance charge is $12 per $100 borrowed up to $400, and $10 per 100 on the amount over $400. What is the dollar amount of interest I am paying? What is the APR of this loan?
9. Using Exhibit 12-1 or the formula, calculate the future value of an ordinary annuity with a quarterly payment of $1,000 made at the end of each quarter for four years compounding quarterly at 8%.
10. Using Exhibit 12-1 or the formula, calculate the future value of an annuity due with a monthly payment of $50 made at the beginning of each month for 2.5 years compounding monthly at 6%.
Answers
Answer:
a) Simple interest = PRT/100 = (800 x 1.5 x 5)/100 = $60
b) Future value = P(1 + RT) = 800(1 + 0.015 x 5) = $870
Total time of loan = 6 months
Phyllis pays $200 two months into the loan, so she still owes $800 for the remaining 4 months.
Simple interest = PRT/100 = (800 x 2 x 4)/100 = $64
Total interest = $64 + $20 = $84
a) Exact interest = PRT/365 = (500 x 4.5 x 90)/36500 = $5.48
b) Future value = P(1 + RT) = 500(1 + 0.045 x 90/365) = $508.22
a) Ordinary interest = PTR/100 = (3500 x 3.5 x 60)/360 = $183.75
b) Proceeds = P - I = 3500 - 183.75 = $3316.25
c) Amount to be paid at maturity = P = $3500
d) Due date = May 4 + 60 days = July 3
APY = (1 + R/n)^n - 1 = (1 + 0.16/4)^4 - 1 = 0.1664 or 16.64%
Future value = P(1 + R/n)^(nt) = 4500(1 + 0.04/2)^(2 x 2) = $5,011.03
P = FV/(1 + R/n)^(nt) = 8000/(1 + 0.06/4)^(4 x 5) = $5,764.44
Interest = (900 x 0.12) + (500 x 0.1) = $132
APR = (132/900) x (365/20) x 100% = 730%
Future value = PMT x ((1 + R/n)^(nt) - 1)/(R/n) = 1000 x ((1 + 0.08/4)^(4 x 4) - 1)/(0.08/4) = $47,490.08
Future value = PMT x ((1 + R/n)^(nt) - 1)/(R/n) x (1 + R/n) = 50 x ((1 + 0.06/12)^(12 x 2.5) - 1)/(0.06/12) x (1 + 0.06/12) = $1,327.10
Use the given conditions to find the exact values of sin(2u), cos(2u), and tan(2u) using the double-angle formula.
Answers
In response to the stated question, we may state that Therefore, the trigonometry exact values of sin(u), cos(u), sin(2u), and cos(2u) are 4/5, 3/5, 24/25, and -7/25, respectively. The exact value of sin(t/2) is 2√5 / 5.
what is trigonometry?The study of the connection between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their possible applications in calculations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle characteristics, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric forms.
Using the given triangle, we can find the values of sin(u), cos(u), and tan(u) as follows:
sin(u) = opposite / hypotenuse = 4 / 5
cos(u) = adjacent / hypotenuse = 3 / 5
tan(u) = opposite / adjacent = 4 / 3
To find the values of sin(2u) and cos(2u), we can use the double angle formulas:
[tex]sin(2u) = 2 sin(u) cos(u)\\cos(2u) = cos^2(u) - sin^2(u)\\sin(2u) = 2 (4/5) (3/5) = 24/25\\cos(2u) = (3/5)^2 - (4/5)^2 = -7/25[/tex]
sin(t/2) = ± [tex]\sqrt((1 - cos(t)) / 2)[/tex]
We need to determine the sign of the square root based on the quadrant in which t/2 lies. Since 7t/2 is in the second quadrant (between pi and 3pi/2), t/2 is in the second quadrant as well (between pi/2 and pi). In the second quadrant, sine is positive and cosine is negative. Therefore, we take the positive square root:
[tex]sin(t/2) = \sqrt((1 - cos(t)) / 2)\\= \sqrt((1 - (-3/5)) / 2)\\= \sqrt(8/10)\\= \sqrt(4/5)\\[/tex]
= 2/√5
= 2√5 / 5
Therefore, the exact values of sin(u), cos(u), sin(2u), and cos(2u) are 4/5, 3/5, 24/25, and -7/25, respectively. The exact value of sin(t/2) is 2√5 / 5.
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a. Multiples of: 7: {_ 6: { LCM:
Answers
The least common multiple (LCM) of 6 and 7 is 42. To find the LCM, we must first list out all the multiples of 6 and 7.
What is number?Number is a mathematical object used to count, measure, and label. It is also commonly used to represent a certain quantity. Numbers are a fundamental part of mathematics, and they can be used in a variety of ways. From basic arithmetic operations to complex equations, numbers are essential in the field of mathematics. Numbers can be used to represent any quantity, such as distance, size, time, or money. Numbers can also be used to represent abstract concepts, such as emotions or thoughts.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84
The LCM is the smallest number that is a multiple of both numbers. In this case, the LCM of 6 and 7 is 42. This can be seen because both 6 and 7 have 42 as a multiple and no other number is smaller.
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The least common multiple (LCM) of 6 and 7 is 42. To find the LCM, we must first list out all the multiples of 6 and 7.
What is number?Number is a mathematical object used to count, measure, and label. It is also commonly used to represent a certain quantity. Numbers are a fundamental part of mathematics, and they can be used in a variety of ways. From basic arithmetic operations to complex equations, numbers are essential in the field of mathematics. Numbers can be used to represent any quantity, such as distance, size, time, or money. Numbers can also be used to represent abstract concepts, such as emotions or thoughts.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84
The LCM is the smallest number that is a multiple of both numbers. In this case, the LCM of 6 and 7 is 42. This can be seen because both 6 and 7 have 42 as a multiple and no other number is smaller.
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Complete questions as follows-
How to find LCM of 6 and 7.
You have been hired t0 design family-friendly see-saw. Your design will featurc uniform board (mass M , length L) that can be moved so that the pivot is distance d from the center of the board. This will allow riders t0 achieve static equilibrium even if they are of different mass_ as most people are _ You have decided that each rider will be positioned so that hishher center of mass will be distance Xoffset from the end of the board when seated as shown. You have selected child of mass m (shown on the right) , and an adult of mass times the mass of the child (shown On the left) to test out your prototype. (a) Derive an expression for the torque applied by the adult rider (on the left) in terms of given quantities and variables available in the palette. Assume counterclockwise is positive_ Xoffct Xottct (b) Derive an expression for the torque applied by the child rider (on the right) in terms of given quantities and variables available in the palette . Assume counterclockwise is positive. Derive an expression for the torque applied by the board in terms of given quantities and variables available in the palette_ Othcexpertta.COm Determine the distance d in terms of n, g and the masses and lengths in the problem. Determine the magnitude of the force exerted on the pivot point by the see-saw while in use in terms of given quantities and variables available in the palette'
Answers
Where F_pivot is the magnitude of the force exerted on the pivot point.
What are perpendicular lines?
Perpendicular lines are lines that intersect at a right angle (90 degrees). In other words, if you draw a line perpendicular to another line, the two lines will form four right angles at the point where they intersect.
(a) The torque applied by the adult rider (on the left) can be calculated as the product of the force applied by the rider and the perpendicular distance from the pivot to the force. Let F_a be the force applied by the adult rider and let r_a be the perpendicular distance from the pivot to the force. Then, the torque applied by the adult rider is:
τ_a = F_a * r_a
The perpendicular distance r_a can be calculated using the Pythagorean theorem:
r_a = sqrt((L/2 - d)^2 + Xoffset^2)
Therefore, the torque applied by the adult rider is:
τ_a = F_a * sqrt((L/2 - d)^2 + Xoffset^2)
(b) Similarly, the torque applied by the child rider (on the right) can be calculated as:
τ_c = F_c * sqrt((L/2 + d)^2 + Xoffset^2)
where F_c is the force applied by the child rider and r_c is the perpendicular distance from the pivot to the force, which can be calculated using the Pythagorean theorem.
(c) The torque applied by the board can be calculated as the sum of the torques applied by the two riders:
τ_b = τ_a + τ_c
Substituting the expressions for τ_a and τ_c, we get:
τ_b = F_a * sqrt((L/2 - d)^2 + Xoffset^2) + F_c * sqrt((L/2 + d)^2 + Xoffset^2)
(d) To determine the distance d in terms of given quantities and variables, we can use the condition for static equilibrium, which requires that the sum of the torques about the pivot point is zero:
τ_a + τ_c = 0
Substituting the expressions for τ_a and τ_c and simplifying, we get:
F_a * (L/2 - d) = F_c * (L/2 + d)
Solving for d, we get:
d = (F_a/F_c - 1) * L/4
Substituting F_a = Mg and F_c = mg, where M is the mass of the adult rider, m is the mass of the child rider, and g is the acceleration due to gravity, we get:
d = (M/m - 1) * L/4
(e) The magnitude of the force exerted on the pivot point by the see-saw while in use can be calculated using the condition for static equilibrium, which requires that the sum of the forces in the vertical direction is zero:
F_a + F_c = Mg + mg
Substituting F_a = Mg and F_c = mg, we get:
F_pivot = Mg + mg
Therefore, where F_pivot is the magnitude of the force exerted on the pivot point.
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Let X and Y be independent random variables, uniformly distributed in the interval [0, 1 Find the CDF and the PDF of X-Y
Answers
Let X and Y be independent random variables, uniformly distributed in the interval [0, 1 ]. The CDF of X - Y is FZ(z) = (1/2)(1+z)^2 for -1 ≤ z ≤ 0, 1 - (1/2)(1-z)^2 for 0 ≤ z ≤ 1, 0 for z < -1 or z > 1. The PDF of X - Y is fZ(z) = z + 1 for -1 < z < 0, 1 - z for 0 < z < 1, 0 otherwise.
To find the CDF of X - Y, we first note that the range of X - Y is [0, 1]. Let Z = X - Y, then:
FZ(z) = P(Z ≤ z) = P(X - Y ≤ z)
We can write this as an integral over the joint distribution of X and Y:
FZ(z) = ∫∫[X - Y ≤ z] fXY(x, y) dx dy
Since X and Y are independent, the joint distribution is simply the product of their marginal distributions:
fXY(x, y) = fX(x) fY(y) = 1 * 1 = 1
for 0 ≤ x, y ≤ 1.
Thus, we have:
FZ(z) = ∫∫[X - Y ≤ z] dx dy
= ∫∫[Y ≤ X - z] dx dy
= ∫0^1 ∫0^(x-z) 1 dy dx + ∫0^1 ∫(x-z)^1 1 dy dx
= ∫0^(1+z) (1-z) dx
= (1/2)(1+z)^2 for -1 ≤ z ≤ 0
= 1 - (1/2)(1-z)^2 for 0 ≤ z ≤ 1
Therefore, the CDF of X - Y is:
FZ(z) =
(1/2)(1+z)^2 for -1 ≤ z ≤ 0
1 - (1/2)(1-z)^2 for 0 ≤ z ≤ 1
0 for z < -1 or z > 1
To find the PDF of X - Y, we differentiate the CDF:
fZ(z) = dFZ(z)/dz =
z + 1 for -1 < z < 0
1 - z for 0 < z < 1
0 otherwise
Therefore, the PDF of X - Y is:
fZ(z) =
z + 1 for -1 < z < 0
1 - z for 0 < z < 1
0 otherwise
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Old Navy is having a 25% off sale on all dresses. Nala sees a dress for 60$. what is the discount rate? What is the sale price of the dress?
Answers
Answer: The sale price of the dress is $45.
Step-by-step explanation:
To find the discount rate, we need to calculate the amount of the discount first. Since the dress is on sale for 25% off, the discount amount is:
Discount = 25% x $60 = $15
The discount rate is then calculated by dividing the discount amount by the original price of the dress and multiplying by 100:
Discount rate = ($15 / $60) x 100% = 25%
To find the sale price of the dress, we need to subtract the discount amount from the original price:
Sale price = $60 - $15 = $45
Therefore, the sale price of the dress is $45.
main Street tea company blends black tea that sells for $3.45 a pound with Earl Gray tea that sells for $2.15 a pound to produce 80 lb of mixture that they sell for $2.75 a pound how much of each kind of tea does the mixture contain rounding to the nearest pound
Answers
36.92 lbs. of the $3.45 tea and 43.08 lbs. of the $2.15 tea are needed.
Let x and y be the amount of tea that sells fo 3.45 and 2.15 a pound respectively:
x+y=80....................eq 1
3.45x+2.15y=2.75(80)......eq 2
:
rewrite eq 1 to x=80-y and plug that value into eq 2
:
3.45(80-y) +2.15y=2.75(80)
:
276-3.45y+2.15y=220
:
-1.3y=56
:
y=43.07 pounds of $2.15 tea
:
28x=80-43.07=36.93 pounds of $3.45 tea
Let a= the pounds of the more expensive tea needed
Let b= the pounds of the less expensive tea needed
(1) a+%2B+b+=+80
(2) 345a+%2B+215b+=+80%2A275 (in cents)
--------------------------
In words, (2) says.
(lbs of 'a' tea x price/lb) + (lbs of 'b' tea x price/lb) =
(lbs of mixture x price/lb of mixture)
-------
Multiply both sides of (1) by 215 and then.
subtract from (2)
345a+%2B+215b+=+80%2A275
-215a+-+215b+=+-80%2A215
130a+=+80%2A60
130a+=+4800
a+=+36.92
and, from (1)
(1) a+%2B+b+=+80
36.92+%2B+b+=+80
b+=+80+-+36.92
b+=+43.08
36.92 lbs. of the $3.45 tea and 43.08 lbs. of the $2.15 tea are needed.
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The mixture contains 34 pounds of black tea and 46 pounds of Earl Gray tea.
What is an algebraic expression?
An algebraic expression is a mathematical phrase that contains variables, constants, and mathematical operations.
Let's denote the amount of black tea in pounds by "x" and the amount of Earl Gray tea in pounds by "y".
Since the total amount of mixture is 80 lb, we have:
x + y = 80 ----(1)
We also know that the mixture sells for $2.75 a pound, so the total revenue from selling 80 lb of mixture is:
80 x $2.75 = $220
On the other hand, the cost of the mixture is the sum of the costs of the black tea and the Earl Gray tea, which is:
3.45x + 2.15y
Since the company wants to make a profit, the revenue must be greater than the cost, so we have:
3.45x + 2.15y < $220
We can simplify this inequality by dividing both sides by 0.1:
34.5x + 21.5y < 2200 ----(2)
Now we have two equations with two unknowns (equations (1) and (2)), which we can solve using substitution or elimination.
Substitution method:
From equation (1), we have:
y = 80 - x
Substituting this into equation (2), we get:
34.5x + 21.5(80 - x) < 2200
Simplifying and solving for x, we get:
x < 34.5
Rounding x to the nearest pound, we get x = 34.
Substituting this value into y = 80 - x, we get y = 46.
Therefore, the mixture contains 34 pounds of black tea and 46 pounds of Earl Gray tea.
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A Triangle has a height that is half of 28 yards and an area of 56 yards^2. What is the length of the base of the trangle?
Answers
The length of the base of the triangle is 8 yards.
What is area of a triangle?The formula for calculating a triangle's area is
A = 1/2 * base * height,
where A represents the triangle's area, base its base length, and height its height measured perpendicularly from base to top.
Given that, height of triangle is half of 28 yards and an area of 56 sq. yards.
The area of triangle is given as:
A = 1/2 * base * height
Substitute the value of h = 1/2 (28) = 14 yards, and area = 56.
56 = 1/2 * base * 14
base = 56(2) / 14
base = 8 yards
Hence, the length of the base of the triangle is 8 yards.
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a delivery truck driver charges a fixed base price of $6 for 2 miles. after 2 miles, he charges an additional $2 for every mile after 6 miles he charges an additional $4 for every mile
Answers
The analysis of the relationship in the question and the graph of the relationship indicates that the cost of the delivery truck between 1 mile and 2 miles is a constant, therefore;
The cost of the delivery truck between 1 mile and 2 miles is constant.What is the graph of a dataset?The graph of a dataset displays the relationship, between the variables in the dataset. It is a visual representation of the connection between the variables.
The fixed base price of the delivery truck for (the first) 2 miles = $6
The additional amount the delivery truck charges after 2 miles = $2 per mile
The additional amount charged by the delivery truck after 6 miles = $4 per mile
The part of the question obtained from a similar question posted on the website, includes;
The description of the cost of the delivery truck between 1 mile and 2 milesThe cost of the delivery truck, between 1 mile and 2 miles based on the graph is an horizontal line.
The horizontal line of a graph, indicates that the relationship between the input and output is a constant, such that the output of the relationship, within the interval of the horizontal line is a constant. The correct option is therefore;
b). The cost of the delivery truck between 1 mile and 2 miles is constant
The possible question options includes;
a) More information is required to determine the cost of the delivery truck between 1 mile and 2 miles
b) The cost of the delivery truck between 1 mile and 2 miles, is constant
c) The cost of the delivery truck is decreasing between 1 mile and 2 miles
d) The cost of the delivery truck is increasing between 1 mile and 2 miles
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Let S be the universal set, where:
S={1,2,3,...,18,19,20}
Let sets A and B be subsets of S, where:
Answers
The elements in the set (A∩B¹) = {2,11,12,} and (B∩A¹) ={5,7,8,9,13,15,16,18}
What is set theory?You should understand that Set theory is a branch of mathematical logic that studies sets, which can be informally described as collections of objects such as numbers, alphabets and variables.
The universal set
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
A = {1,2,6,11,12,19,20}
B = {1,5,6,7,8,9,13,15,16,18,19,20}
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
B¹ = {2.3.4.10.11.12.14.17)
A = {1,2,6,11,12,19,20}
(A∩B¹) = {2,11,12,}
The elements of (B∩A¹) is
S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
A = {1,2,6,11,12,19,20}
A¹ = {3,4,5,7,8,9,10,13,14,15,16,17,18}
B = {1,5,6,7,8,9,13,15,16,18,19,20}
(B∩A¹) = {5,7,8,9,13,15,16,18}
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Which of the following statements must be true based on the diagram below? Select
all that apply. (Diagram is not to scale.)
P
R
N
OOR is a segment bisector.
OOR is a perpendicular bisector.
□ O is the vertex of a pair of congruent angles in the diagram.
OR is the vertex of a pair of congruent angles in the diagram.
□ O is the vertex of a right angle.
None of the above.
Answers
OR is the perpendicular bisector and R is the vertex of a pair of congruent angles in the diagram.
What is perpendicular bisector and a pair of congruent angles ?
A perpendicular bisector is a geometric tool that is commonly used in mathematics and geometry. It is used to divide a line segment into two equal parts and is constructed by first finding the midpoint of the line segment. The perpendicular bisector is then constructed by drawing a line perpendicular to the line segment at its midpoint.
In geometry, two angles are said to be congruent if they have the same measure. This means that they have the same size and shape, even if they are oriented differently. In other words, if you were to place one angle on top of the other, they would perfectly overlap.
Explanation of the true statements:
In the given diagram ,
PR=RN and OR is perpendicular .
Hence, OR is the perpendicular bisector because a perpendicular bisector is a line that passes through the midpoint of a line segment and intersects it at a right angle, dividing the segment into two equal parts.
The vertex of a pair of congruent angles is the point where the two angle bisectors intersect.
Hence, R is the vertex of a pair of congruent angles in the diagram.
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