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In Exploration 3. 1. 1 you found the area under the curve f(t) =

between 1 and 3. What was the approximate area that you came up

with? [Select)

In calculus you will learn more about the significance of this activity. At

what x-value would you stop at to have an area of exactly 1?

[Select]

What is that number called? (Select]

The estimated value of the machine after 4 years when the rate of depreciation dV/dt is inversely proportional to the square of t + 1 is $234,375.

Since the rate of depreciation is inversely proportional to the square of t + 1, we can write:

dV/dt = k / (t + 1)²

where k is the constant of proportionality. We can find k by using the initial value of the machine:

dV/dt = k / (t + 1)² = -100,000 / year when t = 0 (the first year)

Therefore, k = -100,000 * (1²) = -100,000.

To find the value of the machine after 4 years, we need to solve the differential equation:

dV/dt = -100,000 / (t + 1)

We can do this by separating variables and integrating:

∫dV / (V - 500,000) = ∫-100,000 dt / (t + 1)²

ln|V - 500,000| = 100,000 / (t + 1) + C

where C is the constant of integration.

We can find C by using the initial value of the machine:

ln|500,000 - 500,000| = 0 = 100,000 / (0 + 1) + C

Therefore, C = -100,000.

Substituting this value of C, we get:

ln|V - 500,000| = 100,000 / (t + 1) - 100,000

ln|V - 500,000| = -100,000 / (t + 1) + ln|e¹⁰|

ln|V - 500,000| = ln|e¹⁰ / (t + 1)²|

V - 500,000 = [tex]e^{10/(t + 1)²)}[/tex]

V = [tex]e^{10/(t + 1)²)}[/tex] + 500,000

Finally, we can estimate the value of the machine after 4 years by substituting t = 3:

V = [tex]e^{10/(3 + 1)²}[/tex] + 500,000

V ≈ $234,375

Therefore, the correct answer is $234,375.

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The cost of one small crate of apples is $12 and the cost of one large crate of peaches is $8. The cost of 7 small cates of apples and 5 large containers of peach is $126.

The system of equations that describe the question is:

3s + 10l = 116 equation 1

11s + 20l = 292 equation 2

Where:

s = cost of one small crate of apples

l = cost of one large crate of peaches

The elimination method would be used to determine the values of s and l.

Multiply equation 1 by 2

6s + 20l = 232 equation 3

Subtract equation 3 from equation 2:

5s = 60

Divide both sides of the equation by 5

s = 60 / 5

s = $12

Substitute for s in equation 1:

3(12) + 10l = 116

36 + 10l = 116

10l = 116 - 36

10l = 80

l = 80 / 10

l = 8

Cost of 7 small crates of apples and 5 large containers of peaches = (7 x 12) + (8 x 5) = $124

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From the given graph, CE is longer than CD.

The length of the line segment bridging two locations in a plane is known as the distance between the points. d=√((x₂ - x₁)²+ (y₂ - y₁)²) is a common formula to calculate the distance between two points. This equation can be used to calculate the separation between any two locations on an x-y plane or coordinate plane.

Coordinates of E(8,6)

Coordinates of C(6,1)

Coordinates of D(3,-3)

x=8, y=6

x=6, y=1

x=3, y=-3

Distance CE=√{(8-6)² +(6-1)²} = √29

Distance CD=√{(6-3)² +(1+3)²}= √25=5

Therefore, CE is longer than CD.

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Step-by-step explanation:

x is the number.

the equation is

7x + 6 = 3

The work required to pump all the water out of the rectangular swimming pool over the top is approximately 2,323,200 ft-lb.

We have,

To find the work required to pump all the water out of the rectangular swimming pool, we can use the concept of work as the force multiplied by the distance.

First, let's calculate the weight of the water in the pool.

The weight of an object is given by the formula:

Weight = mass x gravitational acceleration

Since the density of water is given as 1 lb/ft³, we need to find the volume of water in the pool.

The volume of the pool is given by the formula:

Volume = length x width x depth

Volume = 50 ft x 30 ft x 6 ft = 9000 ft³

Now, let's calculate the weight of the water:

Weight = density x volume x gravitational acceleration

Weight = 1 lb/ft³ x 9000 ft³ x 32.2 ft/s² ≈ 290,400 lb

To pump all the water out over the top, we need to raise it to the height of the pool, which is 8 ft.

The work required to pump the water out is given by the formula:

Work = weight x height

Work = 290,400 lb x 8 ft = 2,323,200 ft-lb

Therefore,

The work required to pump all the water out of the rectangular swimming pool over the top is approximately 2,323,200 ft-lb.

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0.6990 is value of logarithm .

A logarithm is defined simply.

The logarithm represents the power to which a number must be raised to obtain another number (see Section 3 of this Math Review for more about exponents).

                              As an illustration, the base ten logarithm of 100 is 2, since ten multiplied by two equals 100: log 100 = 2, since 102 = 100. Binary logarithms, which have a base of 2, natural logarithms, which have a base of e 2.71828, and common logarithms with a base of 10 are the four most popular varieties of logarithms.

Now, log0.1= log(1/10) =log (10^-1) =-1 log10

Here log 10= 1 .

 log a = 0.05

log (100a) =  log (100 * 0.05)

                 =  log( 5.00)

                 = log(5)

                = 0.6990

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Answer:

x = 1

Step-by-step explanation:

x + x + x + 30 = 33

3x + 30 = 33

3x + 30 - 30 = 33 - 30

3x = 3

x = 3/3 = 1

The probability is a fraction of 43/205 which is approximately 0.2098.

To calculate the probability of the next person who leaves the lecture having gray hair, we need to know the total number of people who left the lecture, as well as the number of people who have gray hair.

From the given data, we can see that there were a total of 50+40+39+33+43 = 205 people who left the lecture. We also know that there were 43 people who had gray hair.

Therefore, the probability of the next person who leaves the lecture having gray hair is:

P(gray hair) = (number of people with gray hair) / (total number of people who left the lecture)

P(gray hair) = 43/205

P(gray hair) ≈ 0.2098

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Answer:

The given equation is in vertex form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Comparing the given equation with the vertex form, we have a = -3, h = -3 and k = 4.

Since a = -3 < 0, the parabola opens downwards and has a maximum point.

To find the maximum value of y, we need to evaluate y at the x-coordinate of the vertex:

x = -3

y = -3(-3+3)^2 + 4 = 4

Therefore, the parabola y = -3(x+3)2 + 4 contains a maximum point and the maximum value of y is 4.

Hence, the answer is option C

Answer:

C)  Maximum point; 4

Step-by-step explanation:

Given parabola:

[tex]y=-3(x+3)^2+4[/tex]

The given parabola is in vertex form:

[tex]\boxed{y = a(x - h)^2 + k}[/tex]

where:

By comparing the given equation with the vertex form, we can see that:

As a < 0, the parabola opens downwards. Therefore, the vertex of the parabola is a maximum point.

The vertex of the parabola is (h, k) = (-3, 4).

Therefore, the maximum value of y is 4, which occurs at x = -3.

Answer:

This is not my own answer it is a copied one.

If h (x) represents the amount of money spent and x the amount of friends, then we can write it as in a pair as (x, h (x)) Then the pair given is (6.5, $92.25) Here you see a problem, x is 6.5, knowing that x represents the amount of friends, this is a problem because you need to have a whole number ( you can't have a 0.5 of a friend)

The exponential notation of 16384 is 2^14

16384 can be expressed in exponential notation as 2^14, where 2 is the base and 14 is the exponent.

In exponential notation, a number is expressed as a base raised to an exponent, where the base is the number being multiplied repeatedly and the exponent represents the number of times the base is multiplied.

In this case, 2 is being multiplied 14 times to arrive at the value of 16384. Exponential notation is a useful way to represent very large or very small numbers in a concise and standardized format, making it easier to work with and compare values across different scales. It is commonly used in scientific and mathematical contexts.

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According to Chebyshev's theorem, at least 75% of the employees will have commuting times that fall within 2 standard deviations of the mean, or between 58.6 minutes and 68.6 minutes.

Chebyshev's theorem states that for any set of data, regardless of its distribution, a certain percentage of the data lies within a certain number of standard deviations from the mean. Specifically, Chebyshev's theorem states that for any data set, at least 1 – 1/k² of the data values will lie within k standard deviations of the mean, where k is any number greater than 1. If k=2, at least 75% of the data values lie within 2 standard deviations of the mean. If k=3, at least 89% of the data values lie within 3 standard deviations of the mean.

Therefore, for a data set with a mean of 63.6 minutes and a standard deviation of 2.5 minutes, we can use Chebyshev's theorem to determine that at least 75% of the employees will have commuting times that fall within 2 standard deviations of the mean, or between 58.6 minutes and 68.6 minutes.

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Solving a system of equations we can see that he cost of a corn dog is $1.25 and the cost of the fries is $3.50

We can define two variables here:

x = cost of a corn dog.

y = cost of the fries.

With the information in the question we can write two equations, these two equations form a system of equations that we can solve, the system of equations is the one below:

2x + 3y = 13

4x + y = 8.50

We can isolate y in the second equation to get:

y = 8.50 - 4x

Replace it in the other one:

2x + 3*(8.5 - 4x) = 13

2x + 25.5 - 12x = 13

-10x = 13 - 25.5

x = -12.5/-10 = 1.25

Then:

y = 8.5 - 4*1.25 = 3.5

The cost of a corn dog is 1.25 dollars and the cost of the fries is 3.50 dollars.

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The expression for the marginal revenue function is MR(q) = 600r - 7.

The given product's revenue function is R(q) = - 7q + 600qr.

To find an expression for the marginal revenue function, we can use the following steps:

Step 1: Take the first derivative of the revenue function with respect to q to obtain the marginal revenue function MR(q).

Step 2: Simplify the expression for MR(q) to record the final result.

In other words, the marginal revenue function MR(q) is the derivative of the revenue function R(q) with respect to q. Here, R(q) = - 7q + 600qr.

So, we have to differentiate R(q) with respect to q to get MR(q).

The derivative of - 7q with respect to q is - 7.

The derivative of 600qr with respect to q is 600r because the derivative of q with respect to q is 1.

MR(q) = dR(q) / dq

= (d/dq)(- 7q + 600r)

= (- 7) + (600r)

= 600r - 7

The equation that represents the marginal revenue function is MR(q) = 600r - 7.

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The answer of the given question based on the changing the denominator of fraction the answer is the fraction a+3/6-2a can be rewritten with a denominator of 2(a²-9) as (3 + a)/(2(a - 3)).

In mathematics,  formula is  mathematical expression or equation that describes  relationship between two or more variables or quantities. A formula can be used to solve problems or make predictions about particular situation or set of data.

Formulas often involve mathematical symbols and operations, like addition, subtraction, multiplication, division, exponents, and square roots. They may also include variables, which are typically represented by letters, and constants, which are fixed values that do not change.

To change the denominator of the fraction a+3/6-2a to 2(a²-9), we need to factor the denominator of the original fraction and then use algebraic manipulation to rewrite it in the desired form.

First, we can factor the denominator of the original fraction as follows:

6 - 2a = 2(3 - a)

Next, we can rewrite the denominator using the difference of squares formula:

2(a² - 9) = 2(a + 3)(a - 3)

Now, we can use the factored form of the denominator to rewrite the original fraction:

(a + 3)/(6 - 2a) = (a + 3)/(2(3 - a)) = -(a + 3)/(-2(a - 3)) = (3 + a)/(2(a - 3))

Therefore, the fraction a+3/6-2a can be rewritten with a denominator of 2(a²-9) as (3 + a)/(2(a - 3)).

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The hypοtenuse's length is c = 17.

Add the square rοοts οf the οther sides tο find the hypοtenuse. Tο find the shοrter side, subtract the squares οf the οther sides, then take the square rοοt.

Using the Pythagοrean theοrem, we can calculate the length οf the right triangle's missing side:

[tex]a^2 + b^2 = c^2[/tex]

where a, b, and c are the lengths οf the triangle's legs, and c is the length οf the hypοtenuse.

The lengths οf the twο legs are given in this case: a = 8 and b = 15. Sο we can plug the fοllοwing values intο the equatiοn:

[tex]8^2 + 15^2 = c^2[/tex]

[tex]64 + 225 = c^2[/tex]

[tex]289 = c^2[/tex]

When we take the square rοοt οf bοth sides, we get:

[tex]c = \sqrt{(289)} = 17[/tex]

As a result, the hypοtenuse length is c = 17.

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Answer:

Step-by-step explanation: 4(2x-1)+3(2x+5)

Expand the expression to eliminate the brackets

8x-4+6x+15.....Expansion

Now simplify by grouping like terms

8x+6x-4+15

8x+6x=14x

-4+15=11

Therefore simplification=14x+11

The score at the 20th percentile is 27.

To find the score at the 20th percentile of the 29 ages for Academy Award winning best actors, follow the steps below:

Arrange the given ages from smallest to largest.

18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77

Determine the total number of data points

n = 29

Find the rank of the percentile

20th percentile = (20/100) * 29 = 5.8 = 6 (rounded to the nearest whole number).The rank of the percentile is 6.

Use the rank to determine the corresponding data value. The corresponding data value is the value at the 6th position when the data is arranged in ascending order. The score at the 20th percentile is 27.

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In the case of the inequality w < 1, we found that the set of solutions is (-∞, 1), which represents all real numbers less than 1.

The inequality w < 1 means that w is less than 1. To identify all the numbers that satisfy this inequality, we need to look for values of w that are less than 1.

We can continue this process and substitute different values of w in the inequality w < 1 to find more solutions. For instance, if we substitute w = -1, we get -1 < 1, which is also true.

Therefore, -1 is a solution to the inequality w < 1. However, if we substitute w = 2, we get 2 < 1, which is false. This means that 2 is not a solution to the inequality w < 1.

Therefore, the set of all numbers that are solutions to the inequality w < 1 is the set of all real numbers that are less than 1. We can represent this set using interval notation as (-∞, 1), where (-∞) represents all numbers less than negative infinity and 1 represents the upper bound of the interval.

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True. If a counting number with two or more digits remains the same with its digits reversed, then the counting number is a multiple of 11.

When a two-digit number is reversed, it becomes a new number with the digits swapped, e.g., 12 becomes 21. The difference between the original number and the reversed number is obtained by subtracting one from the other. For example, the difference between 12 and 21 is 9. It can be observed that the difference between any two-digit number and its reverse is always a multiple of 9.

Now, let's consider the three-digit number ABC. When this number is reversed, it becomes CBA. The difference between the two is

which is a multiple of 11.

Therefore, if a counting number with two or more digits remains the same with its digits reversed, then the counting number is a multiple of 11.

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Complete Question:

If a counting number with two or more digits remains the same with its digits reversed, then the counting number is a multiple of 11. True/ False.

Answer:

3 seed packets

Step-by-step explanation:

So first, we know that for every 1 seed packet Owen uses, he can grow 2 plants. The ratio is 1:2 (1 = # of seed packets, 2 = # of plants). So now, we need to figure out how many seed packets he needs to have a total of 9 plants. Before we calculate anything, we need to subtract 3 from 9 because it is the total number of plants, and we get 6 plants. To calculate the amount of seed packets Owen needs, we need to get the ration (1:2) and multiply it by 3 on both sides because we need 6 plants. 2 × 3 = 6, and 1 × 3 = 3. The ratio is 3:6. So now we know that Owen needs 3 more seed packets in order to have a total of 9 plants in his backyard. :)

Answer:

Is there supposed to be a joke in this?

Answer:

bro what

Step-by-step explanation:

By answering the presented question, we may conclude that from the given graph we can say that zeros are = (-3,-1 ) and (--5,-9) ; y - intercept is, y = -2x/3 + 15 and vertex are = (-0-1)

Mathematicians use graphs to visually display or chart facts or values in order to express them coherently. A graph point usually represents a connection between two or more items. A graph, a non-linear data structure, is made up of nodes (or vertices) and edges. Glue the nodes, also known as vertices, together. This graph includes V=1, 2, 3, 5, and E=1, 2, 1, 3, 2, 4, and (2.5). (3.5). (4.5). Statistical graphs (bar graphs, pie graphs, line graphs, and so on) are graphical representations of exponential development. a logarithmic graph shaped like a triangle.

from the given graph we can say that

zeros are = (-3,-1 ) and (--5,-9)

y - intercept is, y = -2x/3 + 15

vertex are = (-0-1)

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The answer to the distributive property-based problem -3(-4x + 5) = [?]x - [] is 12x - 15.

A mathematical assertion proving the equality of two expressions is known as an equation. Variables, constants, and mathematical processes like addition, subtraction, multiplication, and division are frequently included. Finding the value of the variable that makes an equation correct is the aim of equation solving. Linear equations, quadratic equations, and exponential equations are just a few of the different ways that equations can be expressed.

given

-3(-4x + 5) = [?]x - [ ]

12x - 15

by distributive property

The answer to the distributive property-based problem -3(-4x + 5) = [?]x - [] is 12x - 15.

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(a)The amount of work needed to stretch the spring from 30 cm to 32 cm is 0.6 J. (b) The distance the spring will be stretched by a 20 N force is 0.03 m.

The formula for the force needed to keep a spring stretched beyond its natural length is F = kx where F is the force, k is the spring constant, and x is the distance from the spring's natural length. The spring constant k is given by the formula: k = (Wd)/x² where W is the work done, d is the distance the spring is stretched from its natural length, and x is the distance from the spring's natural length.

Substituting the values for W, d, and x gives: k = (6 J)/(0.10 m)²

k = 600 N/m

Using the formula F = kx and substituting the values for F and k gives: 20 N = (600 N/m)x

Solving for x gives: x = (20 N)/(600 N/m)

x = 0.0333 m.

Hence, the correct answer is 0.03 m.

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mean is average =9+7+12+13+9+3+5÷7=8.29

median is middle term =9

mode is frequented data=9

rang is the difference between maximum and minimum data=13_3=10

None of the available options match the parabola's equation.

To determine the equation of a parabola, we can utilize the vertex form. Assuming we can read the coordinates (h,k) from the graph, the aim is to utilize the coordinates of its vertex (maximum point, or minimum point), to formulate its equation in the form y=a(xh)2+k, and then to determine the value of the coefficient a.

A parabola's vertex form is given by:

[tex]y = a(x-h)^2 + k[/tex]

where (h,k) is the parabola's vertex.

[tex]y = a(x-5)^2 - 3[/tex]

These values are substituted into the equation to produce:

[tex]-15 = a(2-5)^2 - 3[/tex]

[tex]-15 = 9a - 3[/tex]

[tex]-12 = 9a[/tex]

[tex]a = -4/3[/tex]

[tex]y = (-4/3)(x-5)^2 - 3[/tex]

This equation is expanded and simplified to produce:

[tex]y = (-4/3)x^2 + (32/3)x - 53[/tex]

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The velocity of a particle. P. moving along the x-axis is given by the differentiable function v, where (t) is measured is given by:

A differential function v gives the velocity of a particle P travelling down the x-axis, where v(t) is measured in metres per hour and t is measured in hours. v(t) is a declining function that is continuous. The table below shows several examples of v(t) values.

T [hours]                   0           2      4           7      10

v(t) [meters/hour] 20.3 14.4     10  7.3       5

a) We know that the particle's displacement is the area under the curve v(t). We can calculate the particle's displacement by integrating v(t). Because v(t) is a monotonous (constantly declining) differentiable function, it is also Riemann Integrable. There are now five non-uniform subdivisions:

Partition                     t0   t1 t2 t3 t4

T [hours]                      0           2 4 7 10

v(t) [meters/hour]   20.3 14.4 10 7.3 5

Using Right Riemann sum to approximate the displacement of particle between 0 hr and 10 hr is given by:

[tex]\sum_{n=1}^{4}v(t_n)\Delta t_n=v(t_1)(t_1-t_0)+v(t_2)(t_2-t_1)+v(t_3)(t_3-t_2)+v(t_4)(t_4-t_3) \\=(14.4)(2)+(10)(2)+(7.3)(3)+(5)(3) \\=28.8+20+21.9+15 \\=85.7[/tex]

Therefore, the total displacement between 0 hr and 10 hr is is 85.7 meters.

The estimated position of particle P at time t = 10 hour is 115.7 (= 30 +85.7) meters.

b) Because the function v(t) is decreasing and we are estimating the integral using the Right Riemann sum, the approximation in part(a) underestimates the displacement.

c) A second particle Q also moves along the x-axis so that its velocity is given by :

[tex]V_Q(t)=35\sqrt{t}\cos(0.06t^2)\text{ meters per hour for }0\leq t\leq 10.[/tex]

Hence, the time interval during which the velocity of a particle is atleast 60 meters per hour is [9.404, 10].

Now, the time periods during which a particle's velocity is at least 40 metres per hour are [1.321,4.006] and [9.218, 10]. The distance travelled by the particle Q when its velocity is at least 40 metres per hour is then calculated. :

[tex]\int_{1.321}^{4.006}v_Q(t)dt+\int_{9.218}^{10}v_Q(t)dt\\\\=\int_{1.321}^{4.006}35\sqrt{t}\cos(0.06t^2)dt+\int_{9.218}^{10}35\sqrt{t}\cos(0.06t^2)dt[/tex]

d) At time t = 0, particle Q is is at position x = -90.

We know that P is at xp =  115.7 meters.

Now, The position of Q at t = 10 hr is xq:

[tex]x_q=-90+\int_{0}^{10}v_Q(t)dt=-90+\int_{0}^{10}35\sqrt{t}\cos(0.06t^2)dt[/tex]

And the distance between Q and P is given by :

[tex]|x_p-x_q|=|115.7-(-90+\int_{0}^{10}35\sqrt{t}\cos(0.06t^2)dt)|[/tex]

               [tex]\\=|205.7-\int_{0}^{10}35\sqrt{t}\cos(0.06t^2)dt|[/tex]

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Answer:

4m + 6

Step-by-step explanation:

Since the length is twice as long your equation should look like this

2(2m + 3) = L

which would be 4m + 6 as the length of the rectangle