Answer:
cherry street is 85 yards, while orange dr. and peach ave. is 113 yards, meaning that cherry st. is 28 yards faster
Step-by-step explanation:
85 x 85) - (77 - 77) = 1296
square root of 1296 is 36
77 + 36 = 113
113 - 85 = 28
this means that cherry st. is 28 yds faster
using the Pythagorean Theorum
hope i helped
-lvr
What is the value of y
when 8y/4y +5 = 5/6?
A: 25/7
B: 5/28
C: 15/8
D: 25/28
Answer:
D
Step-by-step explanation:
please mark brainliest thanks:)
A company prepares their shipments in two different-sized boxes.
In order to fit with new shipping regulations, the company needs to decrease the volume of the boxes and will do this by reducing each of the dimensions by at least x inches.
Which system of inequalities can be used to model V, the volume of each box, after each dimension has been reduced by at least x inches?
Answer:
Answer is A. -296x+960/ -636x+3,024
Step-by-step explanation:
Answer:
HEREEE besties
f(x, y) = 2(xo-* yot) _40536 orgs 7.6 Otherwise covariance X 1 J = ?
Covariance is the measure of the joint variability of two random variables. It measures how much two random variables are linearly related to each other. It is denoted by Cov(X, Y).
f(x, y) = 2(xo-* yot) _40536 orgs 7.6, otherwise, covariance X1j = ?The value of covariance X1j can not be determined from the given equation f(x, y) = 2(xo-* yot) _40536 orgs 7.6. There is no relation mentioned between the variables of x and y in the equation. The equation f(x, y) = 2(xo-* yot) _40536 orgs 7.6 means that the function f(x, y) is the multiplication of the value of x to the value of y and then multiplying it by a constant value. The value of the constant is 2(xo-* yot) _40536 orgs 7.6 but the actual value of x and y is unknown. Therefore, we can not calculate the covariance X1j using the given equation. The covariance X1j needs at least two random variables to determine their interdependence, whereas the given equation is a single-variable function that has no relation between two variables .
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Pls help it due soon
Answer:
I need help with math too. Can u help me and I’ll help u
Step-by-step explanation:
$12.60 for 3 boxes. Find the unit rate
Answer:
$4.20 / box
Step-by-step explanation:
12.60 / 3 = 4.20
Which ordered pair is a solution of the equation?
y+1=3(x-4)y+1=3(x−4)y, plus, 1, equals, 3, left parenthesis, x, minus, 4, right parenthesis
Choose 1 answer:
Choose 1 answer:
(Choice A)
A
Only (4,-1)(4,−1)left parenthesis, 4, comma, minus, 1, right parenthesis
(Choice B)
B
Only (5,2)(5,2)left parenthesis, 5, comma, 2, right parenthesis
(Choice C)
C
Both (4,-1)(4,−1)left parenthesis, 4, comma, minus, 1, right parenthesis and (5,2)(5,2)left parenthesis, 5, comma, 2, right parenthesis
(Choice D)
D
Neither
Both (4, 1) and (5, 2) is the solution to the given equation. Therefore, option C is the correct answer.
The given equation is y+1=3(x-4).
A) The given coordinate point is (4, -1).
Substitute (x, y)=(4, -1) in y+1=3(x-4), we get
-1+1=3(4-4)
0=0
So, (4, -1) is the solution to the given equation.
B) The given coordinate point is (5, 2).
Substitute (x, y)=(5, 2) in y+1=3(x-4), we get
2+1=3(5-4)
3=3
So, (5, 2) is the solution to the given equation.
C) Here, both (4, 1) and (5, 2) is the solution to the given equation.
Therefore, option C is the correct answer.
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100 points brainliest!!! plz answer all to the best of your ability no links they don't work for me
Answer the following questions using what you've learned from this unit. Write your responses in the space provided.
1. Part I: The degree of a polynomial is the (greatest / least) of the degrees of its terms. (Circle the term that correctly completes this definition.) (1 point)
Part II: In order to write a polynomial in descending order, you must write the terms with the exponents (decreasing / increasing) from left to right. (Circle the term that correctly completes this rule.) (1 point)
Part III: For each polynomial, determine the degree and write the polynomial in descending order. (4 points: 2 points each)
A. –4x2 – 12 + 11x4 B. 2x5 + 14 – 3x4 + 7x + 3x3
2. Use addition and subtraction to simplify the following polynomials.
A. Add polynomials: (3 – 4x + 8x2) + (–6 + 2x – 5x2)
Step 1: Rewrite the polynomials without the parentheses. (1 point)
Step 2: Write the polynomial in descending order and use parentheses around like terms. (1 point)
Step 3: Add the like terms identified in Step 2 to simplify the polynomial. (1 point)
B. Subtract polynomials: (3x – 5 – 7x2) – (–2 + 6x2 – 5x)
Step 1: Rewrite the polynomials without the parentheses. Remember to multiply each term in the second parentheses by –1. Show your work. (2 points)
Step 2: Write the polynomial in descending order and use parentheses around like terms. (1 point)
Step 3: Add the like terms identified in Step 2 to simplify the polynomial. (1 point)
3. Use the FOIL method to multiply binomials.
Part I: When multiplying binomials, the FOIL method helps you to organize the multiplication of each term of the first binomial by each term of the second. Fill in each blank with the word used to show how the binomials' terms are multiplied. (2 points: 0.5 point each)
F
O
I
L
Part II: Using the FOIL method, multiply the terms in the binomials below. Show your work in the blanks provided. Then, add any like terms and write the polynomial in standard form in the space provided. Show your work. (8 points: 2 points each)
A. (3x + 7)(2x – 5) B. (x2 + 2x)(5x2 – 3x)
______ + _______ + _______ + _______
______ + _______ + _______ + _______
______________________
______________________
C. (5x + 4)(5x – 4) D. (x2 – 7)(x2 – 4)
______ + _______ + _______ + _______
______ + _______ + _______ + _______
______________________
______________________
4. Use the distributive property to multiply the trinomial by the binomial.
Circle the first term in the trinomial, multiply it by each term in the binomial, and place each result in the blank spaces provided. Repeat this process for the second and third terms of the trinomial until all of the spaces are filled in. Finally, in the space provided beneath the blanks, simplify the expression by combining like terms and arranging the terms from highest to lowest order. Show your work. (4 points)
(3x2 – 2x + 7)(x2 + 2x)
______ + _______ + _______ + _______ + ______ + _______
5. Use the vertical method to multiply two trinomials.
Step 1: Multiply the top trinomial by the last term in the bottom trinomial. Place each result in the blank spaces provided. (2 points)
Step 2: Multiply the top trinomial by the middle term in the bottom trinomial. Place each result in the blank spaces provided. (2 points)
Step 3: Multiply the top trinomial by the first term in the bottom trinomial. Place each result in the blank spaces provided. (2 points)
Step 4: Add the three partial products to find the final answer. Place the result at the bottom. (1 point)
6. Sabrina is making an open box from a piece of cardboard that has a width of 12 inches and a length of 18 inches. She'll form the box by making cuts at the corners and folding up the sides. If she wants the box to have a volume of 224 in3, how long should she make the cuts?
Part I: Each dimension (width, length, and height) of the finished box can be represented by a polynomial expression. If the height is x inches, what is the length in terms of x? (2 points)
Height = x
Width = 12 – 2x
Length =
Part II: The formula for volume is v = l • w • h. Use the polynomial expressions from Part I to write an equation to represent the volume of the box. Simplify on the right side by multiplying the polynomials and writing the answer in descending order. Show your work.
HINT: First multiply the binomials representing length and width. Next, multiply the resulting trinomial by the expression representing height. (3 points)
v =
Part III: The length of each cut at the corners is represented by x. To make a box, x must fall in a certain range of values. For example, x cannot be 15 because that would make the flap longer than the width of the box. Identify one reasonable value of x. Why did you choose that value? (2 points)
Part IV: Use the polynomial equation you wrote in Part II to find the volume of the box for each value of x. Show your work. (5 points)
Answer:
Part I: The degree of a polynomial is the greatest of the degrees of its terms.
Part II: In order to write a polynomial in descending order, you must write the terms with the exponents decreasing from left to right.
Part III: A. -4x² + 11x - 12, degree = 2,
B. 5x³ + 4x + 14, degree = 3.
Step-by-step explanation:
Part I : Since the degree of a polynomial is the highest power of its monomials ( single term ),
e: degree of is 5.
Thus, in part I, the correct option is 'greatest'
Part II: When we write a polynomial then we write the terms of the polynomial in descending order of their degrees.
Thus, in part II the correct option is 'least'
Part III: A.
∵ -4x² has the highest power in the polynomial.
⇒ Degree = 2,
Also, in the polynomial descending order of degrees,
2 > 1 > 0
⇒ polynomial in descending order,
B.
Combining like terms,
∵ 5x³ has the highest degree,
⇒ Degree = 3,
Also, the order of the degrees in the polynomial is,
3 < 2 < 1 < 0
Thus, the polynomial in descending order,
Answer: I had trouble with this one too.
Step-by-step explanation:
the length of an arc of a circle is 1/5 of its circumference. if the area of the circle is346.5cm²,find the angle subtended by the arc at the centre of the angle
Answer:
72°
Step-by-step explanation:
From the question,
Area of the circle = πr²
A = πr²................. Equation 1
Where r = radius of the circle.
⇒ r = √(A/π)............. Equation 2
Given: A = 346.5 cm², π = 3.14
r = √(346.5/3.14)
r = √(110.35)
r = 10.5 cm.
Therefore,
circumference of the circle = 2πr = 2×3.14×10.5
circumference = 65.94 m
If the length of the arc(s) is 1/5 of its circumference.
Therefore, length of arc (s) = 13.188
⇒ length of arc/circumference = 13.188/65.94 = 1/5
s/2πr = θ/360
Where θ = angle substends at the center of the circle
1/5 = θ/360
θ = 360/5
θ = 72°
assume x and y are functions of t. evaluate for 2xe^y, with the conditions , x, y0.
The expression to evaluate is 2xe^y, where x and y are functions of t, and the initial conditions are given by x(t=0) = x_0 and y(t=0) = y_0.
To evaluate 2xe^y, we substitute the values of x and y in terms of t into the expression. Since x and y are functions of t, we need to know their specific forms or have additional information about them to proceed with the evaluation.
Without knowing the explicit expressions for x and y or any additional information, we cannot provide a specific numerical evaluation of 2xe^y. However, if you have the specific forms of x(t) and y(t) or any other relevant information, please provide them so that we can assist you further in evaluating the expression.
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For each of these 10 samples, compute Statistics 1, 2, and 3. Enter your answers as 3 numbers separated by commas, such as 1,2,3. 2,3,3* 2,3,4 2, 3, 4* 2, 3+ 4 2, 3*, 4* 2, 4, 4* 3,3*, 4. 3, 3*, 4* 3, 4, 4* I 3*, 4,4* Which statistic would you recommend for estimating ? Average of smallest and largest values in the sample
Statistic 1 would you recommend for estimating average of smallest and largest values in the sample
To compute Statistics 1, 2, and 3 for the given samples, and determine which statistic would be recommended for estimating the average of the smallest and largest values, let's go through each sample one by one:
1. 2,3,3*
Statistics 1: Sum of the values = 2 + 3 + 3 = 8
Statistics 2: Average of the values = (2 + 3 + 3) / 3 = 2.67
Statistics 3: Median of the values = 3
2. 2,3,4
Statistics 1: Sum of the values = 2 + 3 + 4 = 9
Statistics 2: Average of the values = (2 + 3 + 4) / 3 = 3
Statistics 3: Median of the values = 3
3. 2,3,4*
Statistics 1: Sum of the values = 2 + 3 + 4 = 9
Statistics 2: Average of the values = (2 + 3 + 4) / 3 = 3
Statistics 3: Median of the values = 3
4. 2,3*,4*
Statistics 1: Sum of the values = 2 + 3 + 4 = 9
Statistics 2: Average of the values = (2 + 3 + 4) / 3 = 3
Statistics 3: Median of the values = 3
5. 2,4,4*
Statistics 1: Sum of the values = 2 + 4 + 4 = 10
Statistics 2: Average of the values = (2 + 4 + 4) / 3 = 3.33
Statistics 3: Median of the values = 4
6. 3,3*,4
Statistics 1: Sum of the values = 3 + 3 + 4 = 10
Statistics 2: Average of the values = (3 + 3 + 4) / 3 = 3.33
Statistics 3: Median of the values = 3
7. 3,3*,4*
Statistics 1: Sum of the values = 3 + 3 + 4 = 10
Statistics 2: Average of the values = (3 + 3 + 4) / 3 = 3.33
Statistics 3: Median of the values = 3
8. 3,4,4*
Statistics 1: Sum of the values = 3 + 4 + 4 = 11
Statistics 2: Average of the values = (3 + 4 + 4) / 3 = 3.67
Statistics 3: Median of the values = 4
9. 3*,4,4*
Statistics 1: Sum of the values = 3 + 4 + 4 = 11
Statistics 2: Average of the values = (3 + 4 + 4) / 3 = 3.67
Statistics 3: Median of the values = 4
For estimating the average of the smallest and largest values in the sample, the recommended statistic would be Statistics 1, which is the sum of the values. Taking the average of the smallest and largest values can be obtained by dividing the sum by the number of values, which in this case is 3.
Therefore, the recommended statistic for estimating the average of the smallest and largest values is Statistics 1 (sum of the values).
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Mrs. Yellow is an email marketing manager, and she got the following characteristics: 20% open rate, 5% click-through rate, 5% conversion rate, and $1000 average order value. If she was sending the promotion email to a list of 20000 members, her company can achieve (number) of conversions, and $ (number) of revenue." Please write down your answers
The number of conversions Mrs. Yellow's company can achieve is 10, and the revenue they can earn is $10,000.
Mrs. Yellow is an email marketing manager and her company has the following characteristics:
20% open rate, 5% click-through rate, 5% conversion rate, and $1000 average order value. She is sending the promotion email to a list of 20000 members.
To calculate the number of conversions and revenue, we need to use the following formula:
Revenue = Number of conversions × Average order value
Number of conversions = Total number of emails × Conversion rate
Total number of emails = Total number of opened emails × Click-through rate
Total number of opened emails = Total number of sent emails × Open rate
Total number of sent emails = 20,000
Revenue = Number of conversions × Average order value
Total number of sent emails = 20,000
Total number of opened emails = 20,000 × 20/100 = 4,000
Total number of emails = 4,000 × 5/100 = 200
Number of conversions = 200 × 5/100 = 10
Revenue = 10 × 1,000 = $10,000
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Fill in the blank The The hypothesis is a statement that the value of a population parameter is equal to some claimed value
It is a fundamental concept in statistical inference and scientific research.
The hypothesis is a statement that the value of a population parameter is equal to some claimed value. A hypothesis serves as a proposed explanation or prediction about the population based on existing knowledge or observations. In statistical hypothesis testing, two types of hypotheses are considered: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis assumes no significant relationship or difference between variables, while the alternative hypothesis suggests the presence of a significant relationship or difference.
Hypothesis testing involves collecting sample data and performing statistical analysis to evaluate the evidence against the null hypothesis. The goal is to determine whether the data provides enough evidence to reject the null hypothesis in favor of the alternative hypothesis. This process allows researchers to make conclusions about the population based on the available evidence.
Hypothesis testing provides a framework for making objective and informed decisions in various fields of study, such as psychology, biology, economics, and more. It enables researchers to validate or challenge theories, investigate causal relationships, and contribute to the advancement of knowledge in their respective disciplines.
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I will give Brainliest
Emily rides her bike with a constant speed of 18 km/h. How long will she take to travel a distance of 18 kilometers?
Answer:
1 hour
Step-by-step explanation:
Travels 18 km/h
km/h = kilometers per hour
Answer:
1 hour
Step-by-step explanation:
Can someone plz help me with these United rates and explain step by step ?
A new online music service has a collection of 1,500 songs. It expects to triple the number of songs each year for the next few years. Write an equation representing the relationship, where x is time in years, and y is the number of songs available.
Answer: y = (3^(x-1))*1500
Step-by-step explanation:
We want the relationship
y = f(x)
where y is the number of songs, and x is the number of years.
We know that in the first year, x = 1, the site has 1,500 songs.
The next year, x = 2, the number of songs is tripled, then we will have:
1,500*3 = 4500 songs.
The next year, x = 3, we will have: 3*(3*1500) = (3^2)*1500.
when x = 4, the number is tripled again:
3*(3^2)*1500 = (3^3)*1500
We already can see the pattern, for the year x, the site will have:
y = (3^(x-1))*1500 songs.
This is the equation we are looking for.
5 6 7 8 9 + 10 11 12 13 14 15 16 17 18 data a Based on the boxplot above, identify the 5 number summary The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 44 ounces and a standard deviation of 11 ounces. Use the Standard Deviation Rule, also known as the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions. and a) 68% of the widget weights lie between b) What percentage of the widget weights lie between 11 and 55 ounces? S c) What percentage of the widget weights lie above 22?
a. The 68% of the widget weights lie between 33 ounces and 55 ounces.
b. The percentage of widget weights lying between 11 and 55 ounces is approximately 68%.
c. The percentage of widget weights lying above 22 ounces is approximately = 5%.
Based on the given data points: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, the five-number summary can be identified as follows:
Minimum: 5
First Quartile (Q1): 7
Median (Q2): 11
Third Quartile (Q3): 15
Maximum: 18
Now, let's answer the questions related to the distribution of widget weights using the Empirical Rule:
a) The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data lies within one standard deviation of the mean. Since the mean is 44 ounces and the standard deviation is 11 ounces, we can say that approximately 68% of the widget weights lie between 44 - 11 = 33 ounces and 44 + 11 = 55 ounces.
b) To determine the percentage of widget weights lying between 11 and 55 ounces, we need to calculate the z-scores for these values. The z-score is calculated using the formula: z = (x - mean) / standard deviation.
For 11 ounces: z1 = (11 - 44) / 11 = -33/11 = -3
For 55 ounces: z2 = (55 - 44) / 11 = 11/11 = 1
Using the Empirical Rule, we know that approximately 68% of the data lies within one standard deviation of the mean. Therefore, the percentage of widget weights lying between 11 and 55 ounces is approximately 68%.
c) To determine the percentage of widget weights lying above 22 ounces, we need to calculate the z-score for 22 ounces: z = (22 - 44) / 11 = -22/11 = -2.
Using the Empirical Rule, we know that approximately 95% of the data lies within two standard deviations of the mean. Therefore, the percentage of widget weights lying above 22 ounces is approximately 100% - 95% = 5%.
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Please help me I really need it this is hard for me
Answer:
27.6 cm
Step-by-step explanation:
Answer: 44.6
Step-by-step explanation:
It’s not that hard use the formula of the volume of a sphere. And just replace the R with whatever your radius is, if you have a diameter just divide it by 2 to get the radius. Hope this helps
how do you determine a function if a relation is a function
Answer:
Hello! A function is easily identified when an x value does not have more than 1 y value. a y value can have as many x values to infinity, but x can only have one y.
Example...
x y
3 5
4 5
1 2
what is the area of a semicircle that has a radius of 30 feet
Answer:
A = 1413.72 unit²
Step-by-step explanation:
area of a semicircle formula: A = (πr²)/2
To find the area of a semicircle that has a radius of 30 feet, apply r = 30 to the formula A = (πr²)/2:
A = (π30²)/2
A = 1413.72 unit²
A just answer pls I gotta write random words bc the question is a pic
Answer:
$84
Step-by-step explanation:
Is the following equation true? 4m + 12 = 2(m + 6)
Answer:
no
Step-by-step explanation:
4m + 12 ≠ 2m + 12
Check if (1,6) is the solution to the system shown below
Answer:
no
Step-by-step explanation:
To determine if (1, 6) is a solution, substitute x = 1, y = 6 into the left side of both equations and if the value obtained equals the right side then it is a solution.
10(1) - 6 = 10 - 6 = 4 ≠ - 6
- 10(1) + 5(6) = - 10 + 30 = 20 ≠ 30
Then (1, 6 ) is not a solution to the system
7. In multiplying 0.125 x 0.08, Dina knew the rule that says she should multiply 125 x 8 and then move the decimal point 5 places. When she did it on the calculator, she saw it was 0.01. She was confused. How would you help her?
The correct answer to 0.125 x 0.08 is 0.01. Dina's calculator provided the correct result.
The rule Dina mentioned about multiplying 125 x 8 and then moving the decimal point 5 places is not applicable in this case. When multiplying decimal numbers, we need to consider the number of decimal places in each factor.
Here's the correct approach to multiply 0.125 x 0.08:
Step 1: Ignore the decimal point and multiply the numbers as if they were whole numbers:
0.125 x 0.08 = 125 x 8
Step 2: Count the total number of decimal places in the original numbers:
0.125 has three decimal places
0.08 has two decimal places
Step 3: Add the number of decimal places from the original numbers:
Three decimal places + two decimal places = five decimal places
Step 4: Place the decimal point in the result by counting five places from the right:
125 x 8 = 1000
Adding the decimal point five places from the right gives us the final answer: 0.01000
Therefore, the correct answer to 0.125 x 0.08 is 0.01. Dina's calculator provided the correct result.
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Out of her 5 gigabyte data plan, Debbie has used 37%. How much data does she have left?
jawkfjrnsidbjekwbxk2jw dlf 2kqbdkkebakdnrk8w Fflint
Answer:
740%
Step-by-step explanation:
The sale price for a jacket that regularly costs $85.00 is now $68.00. With sales tax, a customer pays $73.44. Which THREE statements are correct for this situation?
Answer:
The sales tax rate is 8%.
C. The sale price is 20% less than the regular price
Step-by-step explanation:
Here are the options to this question :
A) The sales tax rate is 6%
B) The sales tax rate is 8%.
C. The sale price is 20% less than the regular price
Sales tax rate = (tax paid / discounted selling price) x 100
tax paid = amount customer paid - discounted selling price
$73.44 - $68 = $5.44
($5.44 / $68) x 100 = 8%
The sales tax is 8%
Sales price = (discount / initial price) x 100
discount = $85 - $68 = $17
($17 / 85) x 100 = 20%
Answer:
B.) The Sales Tax Rate is 8%
C.) 85-68/85 = discount rate
E.) The Sale Price is 20% less than the regular price.
Step-by-step explanation:
i did the usatestprep :)
The human resource department at a certain company wants to conduct a survey regarding worker benefits. The department has an alphabetical list of all 5705 employees at the company and wants to conduct a systematic sample of size 50.
What is k?
K=
(b) Determine the individuals who will be administered the survey. Randomly select a number from 1 to k. suppose that we randomly select 5.
Starting with the first individual selected, the individuals in the survey will be __ , __, __, __ , __
a) The value of k for the systematic sample is given as follows: k = 114.
b) The individuals are: 1, 2, 3, 4, 5.
What is systematic sample?In a systematic sample, every kth element of the sample out of a sample of n elements is taken.
In this problem, we have a total of 5705 employees, and want a systematic sample of 50 employees, hence the value of k is obtained as follows:
k = 5705/50 = 114.
(rounding the value down to the nearest integer, we just divide the number of people by the sample size to obtain the value of k).
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25 points !! Please help me! What’s the volume to this question? Urgent !
Answer:
381 in^3
Step-by-step explanation:
The volume of the green block is V1 = length(width)(height), or
V1 = (16 in)(7 in)(3 in) = 336 in^3, and
the volume of the red triangular prism is V2 = (1/2)(3 in)(5 in)(6 in) =
V2 = 45 in^3
So the total volume is the sum of these two results: 381 in^3
Most quadratic equations have______roots?
Answer:
equal
Step-by-step explanation:
Most quadratic equations have equal roots.
If incomes increase by 58% and the quantity demanded of tennis balls drops by 87% as a result, what is the income elasticity of demand for tennis balls?
The income elasticity of demand for tennis balls is -1.5. The negative sign indicates an inverse relationship between income and the quantity demanded of tennis balls, suggesting that tennis balls are an inferior good.
To calculate the income elasticity of demand, we need to use the formula:
Income Elasticity of Demand = Percentage change in quantity demanded / Percentage change in income
Given that incomes increase by 58% and the quantity demanded of tennis balls drops by 87%, we can plug these values into the formula:
Income Elasticity of Demand = (-87%)/58%
Simplifying the calculation:
Income Elasticity of Demand = -1.5
The income elasticity of demand for tennis balls is -1.5. The negative sign indicates an inverse relationship between income and the quantity demanded of tennis balls, suggesting that tennis balls are an inferior good.
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Derive the following result "algebraically" using the properties listed in Theorem 6.2.2. Give a reason for every step that exactly justifies what was done in the step For all sets A.B.and C.(AU C-B=(A-B)u(C-B)
we have proven that (A ∪ C) - B = (A - B) ∪ (C - B) algebraically using the properties of set operations
To prove the equality (A ∪ C) - B = (A - B) ∪ (C - B) for all sets A, B, and C, we'll break down the proof step by step, providing a reason for each step based on set operations and properties.
Step 1: Start with the left-hand side (LHS) of the equation: (A ∪ C) - B.
Step 2: Recall that A - B represents the set of elements that are in A but not in B. Similarly, C - B represents the set of elements that are in C but not in B. Therefore, (A ∪ C) - B can be expanded as (A - B) ∪ (C - B).
Step 3: Distribute the union operator (∪) over the set difference (-).
(A ∪ C) - B = (A - B) ∪ (C - B)
Step 4: Apply the distributive property of set operations.
Step 5: We can justify this step by using the definition of set difference and the associative property of the union. For any set X, (X - B) = X ∩ B', where B' represents the complement of set B. Applying this definition to both (A - B) and (C - B), we can rewrite the equation as:
(A ∩ B') ∪ (C ∩ B').
Step 6: Use the distributive property of intersection (∩) over union (∪).
(A ∩ B') ∪ (C ∩ B') = (A ∪ C) ∩ (B' ∪ B')
Step 7: Apply the complement law, which states that B ∪ B' = Universal set (U). Therefore, B ∪ B' = U.
Step 8: Rewrite the equation as (A ∪ C) ∩ U.
Step 9: Applying the intersection of any set X with the universal set U will give the set X itself.
(A ∪ C) ∩ U = A ∪ C.
Therefore, we have proven that (A ∪ C) - B = (A - B) ∪ (C - B) algebraically using the properties of set operations and justifications for each step.
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