we would expect a supervisor rating of approximately 6.61 if a person had a selection test score of 6.
How to find supervisor rating?A mathematical sentence with two equal parts and an equal sign is called an equation. An illustration of an equation is 4 + 6 = 10. 4 plus 6 can be seen on the equal sign's left side, and 10 can be seen on the equal sign's right side.
To find the supervisor rating we would expect for a person with a selection test score of 6, we can plug 6 in for X in the regression line equation:
[tex]$$\hat{Y} = 1.7656 + 0.8073X$$[/tex]
So:
[tex]$$\hat{Y} = 1.7656 + 0.8073(6)$$[/tex]
Simplifying
[tex]$$\hat{Y} = 1.7656 + 4.8438$$[/tex]
[tex]$$\hat{Y} = 6.6094$$[/tex]
Therefore, we would expect a supervisor rating of approximately 6.61 if a person had a selection test score of 6.
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consider the funciton f given by f(t)=log49(t7). find values for a and b such that f(t)=a+logb(t).
The values for a and b such that f(t) = a + logb(t) are,
a = log(1/log(49))
b = 49
Therefor
We can start by simplifying the expression for f(t)
f(t) = log49(t7)
f(t) = log(t7)/log(49) [Using the change of base formula]
Now we can rewrite f(t) as:
f(t) = log(t)/log(b) + a [Using the logarithmic properties log(ab) = log(a) + log(b)]
Comparing this expression with f(t) = a + logb(t), we can see that:
a = log(1/log(49)) [since log(b) = 1/log(49)]
b = t
Therefore, we have
f(t) = log(t7)/log(49) = log(t)/log(b) + log(1/log(49))
= log(t)/log(49) + log(1/log(49))
= log(1/log(49)) + log(t)/log(49)
So, a = log(1/log(49)) and b = 49.
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A delivery truck is transporting boxes of two sizes: large and small. The large boxes weigh 45 pounds each, and the small boxes weigh 35 pounds each. There are 125 boxes in all. If the truck is carrying a total of 4925 pounds in boxes, how many of each type of box is it carrying?
The required number of boxes that the truck contain is 70 small boxes and 55 large boxes.
What is simplification?In mathematics, the operation and interpretation of a function to make it simple or easier to grasp is known as simplifying, and the process is known as simplification.
Given that, Number of large boxes weigh 45 pounds each, and the small boxes weigh 35 pounds each. There are 125 boxes in all, the truck is carrying a total of 4925 pounds
Let the number of large boxes will be l and the number of small boxed be s,
According to the question,
l + s = 125
l = 125 - s - - - - - (1)
Again,
45l + 35s = 4925
put 1 in the above equation,
45[125 - s] + 35s = 4925
5625 - 45s + 35s = 4925
10s = 700
s = 70
Now,
l = 125 - 70
l = 55
Thus, the required number of boxes that the truck contain is 70 small boxes and 55 large boxes.
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A photographer needs a frame for an 5x7 inch picture, such that the total area is 80in^2.Calculate the width of the frame.
Answer:
The width of the frame is 10 inches.
5x7 = 35
80-35 = 45
45/7 = 6.4
5 + 6.4 = 11.4
11.4 rounded down = 11
11 - 5 = 10
Width of the frame = 10 inches
The position of a passenger train that is traveling at an initial speed of 14 feet per second and continues to accelerate can be modeled by the function y = 14t2. A second train that is 1,200 feet ahead of the first train is traveling at a constant speed of 130 feet per second and can be modeled by the function y = 130t + 1200. Solve the system of equations. Which solution represents a viable time that the trains are side by side?
Solving the system of equations, it is found that a viable time that the trains are side by side is given by:
15 seconds
What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
In this problem, the equations are:
[tex]y=14t^2[/tex]
[tex]y=130t+1200[/tex]
They will be side by side when:
[tex]14t^2=130t+1200[/tex]
[tex]14t^2-130t-1200=0[/tex]
It's solution is a quadratic function with coefficients a = 14, b = -130, c = -1200, hence:
[tex]\Delta=(-130)^2-4(14)(-1200)=84100[/tex]
[tex]x_1=\dfrac{130+\sqrt{84100} }{28}=1.5[/tex]
[tex]x_2=\dfrac{130-\sqrt{84100} }{28}=-5.71[/tex]
Time is a non-negative measure, hence 15 seconds is correct.
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what is the area of this circle?
At a community college, a survey was taken to determine where students study on campus. Of the 250 students surveyed, it was determined that
170 studied in the library
135 studied in the cafeteria
76 studied in both the library and the cafeteria
How many studied in library or cafeteria (including both)?
Answer:
Step-by-step explanation:
To find the number of students who studied in the library or cafeteria (including both), we need to add the number of students who studied in the library and the number of students who studied in the cafeteria, but we need to subtract the number of students who studied in both the library and cafeteria to avoid counting them twice.
So, the number of students who studied in library or cafeteria is:
170 + 135 - 76 = 229
Therefore, 229 students studied in the library or cafeteria (including both).
A person invests 10000 dollars in a bank. The bank pays 6.5% interest compounded
annually. To the nearest tenth of a year, how long must the person leave the money
in the bank until it reaches 29500 dollars?
Answer:
Step-by-step explanation:
We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the final amount, P is the principal (initial amount), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).
In this case, we know that P = $10,000, r = 6.5% = 0.065, and we want to find t when A = $29,500. We also know that the interest is compounded annually, so n = 1.
Substituting these values into the formula, we get:
$29,500 = $10,000(1 + 0.065/1)^(1t)
Dividing both sides by $10,000, we get:
2.95 = (1.065)^t
Taking the natural logarithm of both sides, we get:
ln(2.95) = ln(1.065)^t
Using the property of logarithms that ln(a^b) = b ln(a), we can rewrite the right side as:
ln(2.95) = t ln(1.065)
Dividing both sides by ln(1.065), we get:
t = ln(2.95)/ln(1.065) ≈ 18.2
Therefore, the person must leave the money in the bank for about 18.2 years to reach $29,500. To the nearest tenth of a year, the answer is 18.2 years.
Plsss help due today!!
The writer’s guild only gives a lifetime achievement award out every 5 years and was curious to see how the population in Europe felt about their process.
They decided to conduct an opinion survey by selecting a random sample of 500 Europeans who made book purchases online in 2022. They emailed the survey to all 500 randomly selected online shoppers and received 127 responses.
9) Describe the population of interest.,
10) Describe the sample.
11) Give an example of a parameter they might be interested in calculating from the survey.
12) Give an example of a statistic they might report based on the parameter you gave above.
Answer:
The population of interest is all Europeans who make book purchases online in 2022.The sample is the randomly selected 500 Europeans who make book purchases online in 2022, to whom the survey was emailed, and the 127 who responded to the survey.A parameter they might be interested in calculating from the survey is the proportion of all Europeans who make book purchases online in 2022 who support the writer’s guild's process of giving a lifetime achievement award out every 5 years.A statistic they might report based on the parameter above is the proportion of the 127 respondents who support the writer’s guild's process of giving a lifetime achievement award out every 5 years, and an estimate of the proportion of all Europeans who make book purchases online in 2022 who support this process, based on the sample data.PLEASE HELP!!!!!!!!!
△DEF is similar to △YZX
A) which side corresponds to ED? (this one is already answered)
B) write a proportion that you could use to find XZ
C) what is XZ?
Step-by-step explanation:
B) 20:23
C) 6,0375
hope this helps
What is the slope of the following line?
A. 1/2
B. 2
C. -2
D. -1/2
Answer:
D. -1/2
Step-by-step explanation:
slope = rise/run = 1/-2 = -1/2
1. How many quilts will Sally supply at $500.00 per quilt?
2. Are any other factors besides price taken into account with the market supply schedule she has constructed?
3. Several situations are listed below that Sally has encountered in her business. Explain how each of these situations might
alter the supply of quilts.
a new sewing machine that helps her produce quilts in less time.
Sally will supply 3 quilts at $500.00 per quilt according to the market supply schedule she constructed.
What is the supply about?The market supply schedule only takes into account the price per quilt and the quantity supplied per month. Other factors such as production costs, availability of raw materials, competition, and changes in technology are not taken into account in this supply schedule.
a. If Sally purchases a new sewing machine that helps her produce quilts in less time, her production efficiency will increase. This will enable her to produce more quilts in a shorter period of time, which will result in an increase in the supply of quilts.
b. If Sally anticipates an increase in prices for quilts in three months, she may increase the quantity supplied per month in order to take advantage of the higher prices. This may result in a temporary increase in the supply of quilts.
Lastly, c. If there are twice as many quilt suppliers as there were one year ago, Sally will likely face increased competition. This may result in a decrease in sales for Sally and a decrease in the price of quilts. In order to remain competitive, Sally may need to reduce the quantity supplied per month or find ways to differentiate her quilts from those of her competitors.
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See full question below
Part One
Sally Thomas is an entrepreneur. She makes hand-made quilts and sells them at craft shows and at her home studio. Below is a market supply schedule she constructed. Answer the questions that appear below it.
Price per quilt
Quantity Supplied Per Month
$300.00
1
$400.00
2
$500.00
3
$600.00
4
$700.00
5
1. How many quilts will Sally supply at $500.00 per quilt?
3
2. Are any other factors besides price taken into account with the market supply schedule she has constructed?
Supply
3. Several situations are listed below that Sally has encountered in her business. Explain how each of these situations might alter the supply of quilts. a. Sally purchases a new sewing machine that helps her produce quilts in less time.
Sally will be able to produce more quilts in less time, increasing the supply of quilts.
b. Sally anticipates an increase in prices for quilts in three months.
Sally will increase the quantity supplied per month for three months.
c. There are twice as many quilt suppliers as there were one year ago.
Sally will likely experience a decrease in sales which in turn will result in a decrease of supply and price.
Last years freshman class at big state university totaled 5,303 students
URGENT
The 1,262 students who received a merit scholarship, the amount they received varied per student and totaled an average of $3,458 ($454). That amount is 78.2% of the full tuition cost of $4,400.
What is merit?Merit is a term used to describe the quality of something or someone that makes them worthy of recognition or respect. It is an indicator of worthiness and is usually based on a person's ability, effort, or accomplishments. Merit is often used when evaluating an individual or a group for a promotion, hiring, or award. Merit is subjective, as different people have different standards for what merits recognition.
Therefore, 21.8% of the students who received a merit scholarship did not receive enough to cover full tuition. Therefore, the percentage of students who received a merit scholarship and did not receive enough to cover full tuition is 21%.
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If a + b = 3 and ab = 2, find the value of a² + b²
Answer:
Through observation, a=2, b=1 or vice versa .
Step-by-step explanation:
If a=2 and b=1
a² +b²=?
2² +1² =4+1=5
a²+b²=5
let a and b be positive integers, where a and b do not equal 0. for what limit expression does l'hospital's rule apply? (1 point)
For a limit expression to be solved using L'Hospital's rule, the expression must be in the form of 0/0 or infinity/infinity, and the derivatives of the numerator and denominator must exist and approach a finite limit or infinity/negative infinity at the point of evaluation.
L'Hospital's rule applies to limit expressions of the form 0/0 or infinity/infinity. Specifically, if we have a limit expression of form f(x)/g(x), where f(x) and g(x) both approach 0 or infinity as x approaches a certain value, then L'Hospital's rule may be applied to find the limit of the expression.
It is important to note that L'Hospital's rule can only be applied if the limit of the derivative of f(x) divided by the derivative of g(x) exists and is finite, or if the limit of the derivative of f(x) divided by the derivative of g(x) approaches infinity or negative infinity.
Therefore, for a limit expression to be solved using L'Hospital's rule, the expression must be in the form of 0/0 or infinity/infinity, and the derivatives of the numerator and denominator must exist and approach a finite limit or infinity/negative infinity at the point of evaluation.
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Victor has been saving up money to buy the new iPhone 14 Pro Max , which retails for $ 1099 . He currently has from Lunar New Year , and started working as a tutor for $ 50 a week . The amount of money he has in his savings account can be modeled by V(w) = 50w + 500 where V(w) is the amount he has for w weeks of working as a tuto State the value of the slope and y - intercept . a b . What is the meaning of the slope and y - intercept ? How many weeks will he have to work to buy the new iPhone 14 Pro Max ?
For Victor to have enough money to purchase the new iPhone 14 Pro Max, he will need to work for roughly 12 weeks.
Is it an equation or an expression?An expression is made up of a number, a variable, or a combination of a number, a variable, and operation symbols. An equation is created by joining two expressions with an equal sign. For illustration: When you add 8 and 3, you get 11.
a) Here is the equation for Victor's savings account:
V(w) = 50w + 500
Slope (m) = 50
Y-intercept (b) = 500
b) The slope (m = 50) represents the rate at which Victor is saving money per week. For every week that he works as a tutor, he saves $50. The y-intercept (b = 500) represents the initial amount of money that Victor had saved up before he started working as a tutor.
(c)To find out how many weeks Victor will have to work to buy the new iPhone 14 Pro Max, we need to set up an equation where the amount of money he has saved up (V(w)) is equal to the price of the phone ($1099). Therefore, we can write:
50w + 500 = 1099
Subtracting 500 from both sides, we get:
50w = 599
Dividing both sides by 50, we get:
w = 11.98
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A bag with 12 marbles is shown below. (4 marbles are blue, 3 are red, and 5 are yellow.) A marble is chosen from the bag at random. What is the probability
that it is blue?
Write your answer as a fraction or a whole number.
The value of the probability of selecting a blue marble from the bag is 1/3
Calculating the probability of selecting a blue marbleThe probability of selecting a blue marble can be found by dividing the number of blue marbles by the total number of marbles in the bag.
From the given information, we know that there are 4 blue marbles out of a total of 12 marbles in the bag.
Therefore, the probability of selecting a blue marble is:
Probability of selecting a blue marble = Number of blue marbles / Total number of marbles
This gives
Probability of selecting a blue marble = 4/12
Simplifying the fraction by dividing both the numerator and denominator by 4, we get:
Probability of selecting a blue marble = 1/3
So, the probability of selecting a blue marble from the bag is 1/3 or 0.33 as a decimal.
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sabina needs to make a cake and some cookies. The cake requires 3/8 cup of sugar and the·cookies require 3/5 cup of sugar. Sabina has 15/16 cup of sugar. Does she have enough sugar,or how much more does she need
Answer:
No
She needs 3/80 cup more
Step-by-step explanation:
She needs 3/8 cup plus 3/5 cup.
3/8 + 3/5 = 15/40 + 24/40 = 39/40
She has 15/16 cup
15/16
39/40 = 78/80
15/16 = 75/80
She has 75/80 cup and needs 78/80 cup.
She does not have enough sugar.
78/80 - 75/80 = 3/80
She needs 3/80 cup more.
A machine can dig a land of 5 bighas in 14 hours. How long will it take to dig 60 bighas of land by the same machine?
Answer168 Bighas of land.
Step-by-step explanation:
Since the machine can dig a land of five bighas in 14 hours, put it into a ration 5:14 5(land it can dig) 14(amount of hours it takes.) Have X as the time to dig 60 bighas, 60(amount of land) : x (time). The equation would be 60/x=5/14 ===> 168 bughas of land.
Using a standard deck of cards, a gamer drew one card and recorded its value. They continued this for a total of 100 draws. The table shows the frequency of each card drawn.
Card A 2 3 4 5 6 7 8 9 10 J Q K
Frequency 4 7 5 6 7 6 8 10 7 10 8 12 10
Based on the table, what is the experimental probability that the card selected was a 3, 5, or 7?
one fifth
one third
3 over 10
6 over 13
This can be simplified to 19/100, which is equivalent to 6/31 or approximately 0.19 or 19%. The answer is: 6 over 31.
what is probability?
Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 indicates that an event is impossible, and 1 indicates that an event is certain. Probability can be expressed as a fraction, decimal, or percentage.
The question asks for the experimental probability of drawing a 3, 5, or 7 from a deck of cards, given the frequency table provided.
The frequency table tells us how many times each card was drawn out of a total of 100 draws.
To calculate the frequency of drawing a 3, 5, or 7, we add the frequencies of these cards: 4 (for 3) + 6 (for 5) + 6 (for 7) = 16. Note that we do not include the frequencies of any other cards.
To calculate the probability of drawing a 3, 5, or 7, we divide the frequency of these cards (16) by the total number of draws (100): 16/100 = 0.16. This means that in our experiment, we drew a 3, 5, or 7 approximately 16% of the time.
The total number of draws is 100, and the frequency of cards 3, 5, and 7 is 7+6+6=19.
Therefore, the experimental probability of drawing a 3, 5, or 7 is:
19/100 = 0.19
This can be simplified to 19/100, which is equivalent to 6/31 or approximately 0.19 or 19%.
Therefore, the answer is: 6 over 31.
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What is the value of “a”
What is the vertex form of the graph
what is the standard form of the graph
what is the factored form of the graph
The value of a is 2.
the vertex form of the graph : y = 2 ( x + 1 )² - 9,
the standard form of the graph : y = 2x² + 4x - 7,
the factored form of the graph : y = 2 ( x + 1) ( x + 1 ) - 9.
What is the quadratic equation?
The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.
The equation of the parabola in vertex form:
The vertex v is at v ( -1, -9 ) this is equivalent to v ( h, k )
the equation of parabola in vertex form is y = a ( x- h )² + k
y = a ( x - (-1) )² + (-9)
substituting point ( -3, -5 ) from the graph we have:
-5 = a ( -3 + 1 )² - 9
-5 = 2a - 9
a = 2
substituting the known values to y = a ( x- h ) + k
y = 2 ( x + 1 )² - 9
the equation of the parabola in factored form
y = 2 ( x + 1) ( x + 1 ) - 9
the equation of the parabola in standard form
y = 2 ( x + 1) ( x + 1 ) - 9
y = 2 ( x² + 2x + 1 ) - 9
y = 2x² + 4x - 7
Hence, the value of a is 2
the vertex form of the graph : y = 2 ( x + 1 )² - 9
the standard form of the graph : y = 2x² + 4x - 7
the factored form of the graph : y = 2 ( x + 1) ( x + 1 ) - 9
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4.1 h(x) Consider h(c) = cos 2x 4.1.1 Complete the table below, rounding your answer off to the first decimal where needed: -90° -75° -60° -45° -30° -15° 0° 15° 30° 45° 60° 75° 90° 4.1.2 Now use the table and draw the graph of h(x) = cos 2x on the system of axes below: -90°-75°-60-45-30-15 2- 14 - 1+ -24 (2) 15° 30° 45° 60⁰ 75⁰ 90⁰ (2) (2)
Here's the completed table and the graph:
x h(x)
-90° 1.0
-75° -0.5
-60° -1.0
-45° -0.0
-30° 1.0
-15° 0.5
0° 1.0
15° 0.5
30° -0.0
45° -1.0
60° -0.5
75° 1.0
90° 1.0
What is function?In mathematics, a function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are often represented as a set of ordered pairs, where the first element of each pair is an input and the second element is the corresponding output. Functions are a fundamental concept in many areas of mathematics and have many real-world applications, including in science, engineering, and economics.
Here,
To calculate the values of h(c) in the table, we plug in the given values of x into the function h(c) = cos 2x and evaluate. For example, to find h(c) when x = -75°:
h(c) = cos 2x
h(c) = cos 2(-75°) (substitute -75° for x)
h(c) = cos (-150°) (simplify using the double angle identity)
h(c) = -0.5 (evaluate using the unit circle or a calculator)
We repeat this process for each value of x to fill out the table.
To graph the function h(x) = cos 2x, we plot each point from the table on the given system of axes. The x-axis represents the angle x in degrees, and the y-axis represents the value of h(x) = cos 2x. We then connect the points with a smooth curve to obtain the graph.
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Identify the sampling method as simple random sampling, systematic sampling, convenience sampling, or stratified sampling. A computer randomly selects 100 names from a list of all registered voters. Those selected are surveyed to predict who will win the election for Mayor. Choose the correct sampling technique below. A. Systematic sampling B. Convenience sampling c. Simple random sampling D. Stratified sampling
The correct sampling technique for the given case is C. Simple random sampling
Each person in a population has an equal probability of being chosen to be a member of specific sample when using simple random sampling as a statistical sampling technique. With this technique, a random sample is picked at random from a wider population, giving each person a fair chance of being chosen. This indicates that there is no bias in selection process and that any person or object in overall population has same likelihood of getting chosen.
Simple random sampling is frequently employed in research investigations because it is an easy-to-use and objective sampling technique. Given that 100 names were randomly chosen from a list of all registered voters, simple random sampling is appropriate sampling technique in this particular case.
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Question 6 of 10
Based only on the information given in the diagram, which congruence
theorems or postulates could be given as reasons why ACDE AOPQ?
Check all that apply.
AA
A. AAS
B. ASA
C. LL
OD. HL
E. LA
F. SAS
Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.
what is triangle ?Having three straight sides and three angles where they intersect, a triangle is a closed, two-dimensional shape. It is one of the fundamental geometric shapes and has a number of characteristics that can be used to study and resolve issues that pertain to it. The triangle inequality theory states that the sum of a triangle's interior angles is always 180 degrees, and that the longest side is always the side across from the largest angle. Triangles can be used to solve a wide range of mathematical issues in a variety of disciplines and can be categorised based on the length of their sides and the measurement of their angles.
given
We can use the following congruence theories or postulates based on the data in the diagram:
A. ASA
B. AAS
C. LL (corresponding angles hypothesis)
F. SAS
Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.
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The current population of a city is 8.35 × 105 people and the city is expected to grow in population by approximately 1.6 x 10^ people next year
Answer:
Step-by-step explanation:
8.35 times 105 is 876.75
1.6 times 10 is 16
Hope this helps
Find the coefficient! Answer the following questions with the assistance of the Binomial Theorem
(Theorem 17.8):
a. What is the coefficient of .x 3 in (1 + .x 6?
b. What is the coefficient of .x 3 in (2.x - 3 6 ?
"c. What is the coefficient of .x 3 in (x + 1 20 + (.""\: - I )20 ?"
e. What is the coefficient of .x 3 y 3 in (.x + 1?
a. Coefficient of .x³in (1 + .x)⁶ is 20. b. Coefficient of .x³ in (2.x - 3)⁶ is -540. c. Coefficient of .x³ in (x + 1)²⁰ + (x - 1)²⁰ is zero. d. Coefficient of .x³ y³ in (.x + 1)⁴ is 4.
a. To find the coefficient of .x³ in the expansion of (1 + .x)⁶, we can use the binomial theorem. The coefficient of .x³ is given by the expression 6C₃(1)³(.x)³, where 6C₃ is the number of ways to choose three items from a set of six items. Evaluating this expression gives us:
6C₃(1)³(.x)³ = (6!)/(3!3!)(1³)(.x)³ = 20(.x)³
Therefore, the coefficient of .x³ in (1 + .x)⁶ is 20.
b. Similarly, we can use the binomial theorem to find the coefficient of .x³ in the expansion of (2.x - 3)⁶. The coefficient of .x³ is given by the expression 6C₃(2.x)³(-3)⁶, which simplifies to:
6C₃(2³)(.x)³(-3)⁶ = 6C₃(8)(.x)³(729)
The value of 6C₃ is 20, so we have:
20(8)(.x)³(729) = 116,640(.x)³
Therefore, the coefficient of .x³ in (2.x - 3)⁶ is -116,640.
c. In the expansion of (x + 1)²⁰ and (x - 1)²⁰, each term will be of the form (xⁿ)(1)⁽²⁰⁻ⁿ⁾ and (xⁿ)(-1)⁽²⁰⁻ⁿ⁾, respectively, where n is an even integer between 0 and 20. Since the powers of x are even, the only term that can contain .x³ is the term where n = 6, which is (x⁶)(1⁽¹⁴⁾) and (x⁶)(-1⁽¹⁴⁾) in the two expansions. Adding these two terms together gives us:
(x⁶)(1⁽¹⁴⁾ - 1⁽¹⁴⁾) = 0
Therefore, the coefficient of .x³ in (x + 1)²⁰ + (x - 1)²⁰ is zero.
d. To find the coefficient of .x³ y³ in (.x + 1)⁴, we can use the binomial theorem once again. The coefficient of .x³ y³ is given by the expression 4C₁(1³)(.x)³(y)¹, which simplifies to:
4C₁(1³)(.x)³(y)¹ = 4(.x)³(y)
Therefore, the coefficient of .x³ y³ in (.x + 1)⁴ is 4.
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A mass that weighs 8 lb stretches a spring 24 in. The system is acted on by an external force of 4sin4t. If the mass is pulled down 6 in and then released, determine the position of the mass at any time t. (Use g=32ft/s2 for the acceleration due to gravity. Let u(t), measured positive downward, denote the displacement in feet of the mass from its equilibrium position at time t seconds.)
The position of the mass at any time t is given by u(t) = -1/2 cos(2√2 t) + sin(2√2 t) + 1/2 sin(4t) - 1/2 cos(4t), and the first four times at which the velocity of the mass is zero, excluding t = 0, are t = 0.0567, 0.283, 0.510, and 0.736.
To solve this problem, we need to use Newton's second law of motion which states that the force acting on an object is equal to its mass times its acceleration. In this case, the force acting on the mass is the sum of the force due to gravity and the external force, and the acceleration is given by the second derivative of the displacement.
Using Hooke's law, the force due to the spring is proportional to the displacement u of the mass from its equilibrium position, with a proportionality constant k
F_spring = -k u
where k is the spring constant. Since the mass is pulled down 6 in. from its equilibrium position, the initial displacement is u(0) = -0.5 ft. The spring stretches 24 in. under a weight of 8 lb, so the spring constant is
k = F_spring / u = (8 lb) / (2 ft) = 4 lb/ft
The force due to gravity is
F_gravity = m g
where m is the mass of the object and g is the acceleration due to gravity. The mass is given in pounds, so we need to convert it to slugs:
m = 8 lb / (32.2 ft/s^2) = 0.248 slugs
Thus, the force due to gravity is
F_gravity = (0.248 slugs) (32 ft/s^2) = 7.94 lb
The equation of motion for the displacement u(t) is
m u''(t) = F_gravity + F_spring + F_external
where F_external = 4 sin 4t lb is the external force. Substituting the expressions for these forces and dividing by m, we get
u''(t) + 8 u(t) = 2 sin 4t
This is a homogeneous linear second-order differential equation with constant coefficients, which can be solved by finding the roots of the characteristic equation:
r^2 + 8 = 0
The roots are r = ±2i√2. The general solution of the differential equation is then
u(t) = c1 cos(2√2 t) + c2 sin(2√2 t) + u_p(t)
where u_p(t) is a particular solution of the nonhomogeneous equation. Since the right-hand side of the equation is a sinusoidal function with frequency 4/π, we can guess a particular solution of the form
u_p(t) = A sin(4t) + B cos(4t)
Substituting this into the equation and equating coefficients, we get
16 A - 16 B + 8 A = 2, or A = 1/2
-16 A - 16 B + 8 B = 0, or B = -1/2
Therefore, the particular solution is
u_p(t) = 1/2 sin(4t) - 1/2 cos(4t)
The general solution is then
u(t) = c1 cos(2√2 t) + c2 sin(2√2 t) + 1/2 sin(4t) - 1/2 cos(4t)
We can use the initial conditions u(0) = -0.5 and u'(0) = 0 to determine the values of c1 and c2
u(0) = c1 + 1/2 = -0.5, or c1 = -1/2
u'(0) = -2√2 c1 + 2√2 c2 - 2 = 0, or c2 = 1
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The given question is incomplete, the complete question is:
A mass that weighs 8 lb stretches a spring 24 in. The system is acted on by an external force of 4 sin 4t. If the mass is pulled down 6 in. and then released, determine the position of the mass at any time t. (Use g = 32 ft/s2 for the acceleration due to gravity. Let u(t), measured positive downward, denote the displacement in feet of the mass from its equilibrium position at time t seconds.)
u(t) = 1/2sin(4t)+1/2cos(4t)-2tcos(4)
Determine the first four times at which the velocity of the mass is zero. (Exclude t = 0 as trivial.)
Bradley went to the store to buy ingredients for a new recipe. Artichokes were on sale for $3 per pound.
How much did Bradley pay if he bought
2
3
of a pound?
A $6. B $5. C $3 D $2
Answer :
Step-by-step explanation to problem:
2/3 * 3 = 2
we do 2/3 times 3 because $3 is for 1 pound and here we only need 2/3 of a pound
$2
Correct Answer = D
Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x) = x^3 - x^2 - 37x - 35 Find the real zeros of f. Select the correct choice below and; if necessary, fill in the answer box to complete your answer. (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression. Use a comma to separate answers as needed.) There are no real zeros. Use the real zeros to factor f. f(x)= (Simplify your answer. Type your answer in factored form. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression.)
By using rational zeros theorem, we find that there are no real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, so we cannot factor f(x) over the real numbers.
To find the real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, we can use the rational zeros theorem, which states that any rational zeros of the function must have the form p/q, where p is a factor of the constant term (-35) and q is a factor of the leading coefficient (1).
The possible rational zeros of f are therefore ±1, ±5, ±7, ±35. We can then test each of these values using synthetic division or long division to see if they are zeros of the function. After testing all of the possible rational zeros, we find that none of them are actually zeros of the function.
Therefore, we can conclude that there are no real zeros of the function f(x) = x^3 - x^2 - 37x - 35.
However, we could factor it into linear and quadratic factors with complex coefficients using the complex zeros of f(x). But since the problem only asks for factoring over the real numbers, we can conclude that the factored form of f(x) is:
f(x) = x^3 - x^2 - 37x - 35 (cannot be factored over the real numbers)
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Emma is choosing a weekly meeting time. She hopes to have two different
managers attend on a regular basis. The table shows the probabilities that
the managers can attend on the days she is proposing.
Manager A
Manager B
Monday
0.82
0.88
Wednesday
0.87
0.85
Assuming that manager A's availability is independent of manager B's
availability, which day should Emma choose to maximize the probability that
both managers will be available?
On Wednesday, there is a higher (0.7395) likelihood that both managers will be accessible than on Monday (0.7216).
what is probability ?The study of random occurrences and their results falls under the category of probability, which is a subfield of mathematics. A number between 0 and 1, where 0 denotes an impossibility and 1 denotes a certainty, is used to express the likelihood of an event happening. By dividing the number of favourable outcomes by the total number of potential outcomes, the probability of an occurrence is determined. It is used in many fields, including statistics, finance, engineering, and science, to generate predictions and inform decision-making.
given
We must calculate the product of the odds of each manager being available on a given day in order to determine the day with the highest likelihood that both managers will be available.
The likelihood that both managers will be accessible on Monday is 0.82 x 0.88, or 0.7216.
The likelihood that both managers will be accessible on Wednesday is 0.87 x 0.85 = 0.7395.
Therefore, the solution is option C: Wednesday. On Wednesday, there is a higher (0.7395) likelihood that both managers will be accessible than on Monday (0.7216).
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All of the angles in the pentagon below are equal. What is the measure of each interior angle in the pentagon below?
Step-by-step explanation:
Here is ONE way
sum of exterior angles of a polygon = 360 °
there are 5 , se each EXTERIOR angle = 360/5 = 72 °
the EXT angles are SUPPLEMENTARY with the interior angles
ext + int = 180°
72 + int = 180° INTERIOR angle = 108 °