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Business Calculus Functions

Given the following table...(a) Is y a function of x? Explain your answer. (b)Is x a function of y? Explain your answer.
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Hi, I am having trouble doing these problems listed below. Please show me how to solve these problems for future reference. Thank you very much.

I would like for you to show me all of your work/calculations and the correct answer to each problem.

1. Given the following table:

x 2 4 8 9 10 11 10 15
y 4 5 7 9 11 13 15 17

(a) Is y a function of x? Explain your answer.
(b) Is x a function of y? Explain your answer.

Answers:

1. (a) No, y is not a function of x because there is not exactly one y value for each x value.
(b) Yes, x is a function of y because there is exactly one x value for each y value.

2. Given the following table:

x 1.22 1.43 1.08 1.99 1.51 1.13 1.51 1.72
y 4 5 7 9 10 11 9 15

(a) Is y a function of x? Explain your answer.
(b) Is b a function of y? Explain your answer.

13. A basketball team keeps a record of the number of points scored by the star player in each game:

Game 1  23 points
Game 2  18 points
Game 3  21 points
Game 4  18 points
Game 5  19 points
Game 6  22 points
Game 7  17 points
Game 8  24 points

(a) Is the number of points scored a function of the game number? Why or why not?
(b) Is the game number a function of the number of points scored? Why or why not?

Answers:

(a) Yes. The number of points is a function of the game because there is exactly one point value for each game.
(b) No. The game number is not a function of the points scored because the value of 18 points could come from Game 2 or 4.

14. The table below gives the results of a survey showing the price charged for a sweatshirt and the number of students saying that the price is the most they would pay, to the nearest $2.50.

Price ($) 0 2.50 5.00 7.50 10 12.50 15 17.50 20 22.50 25

# of Students 9 3 2 12 15 18 17 12 8 3 1

(a) Is the number of students a function of the price? Why or why not?
(b) Is the price a function of the number of students? Why or why not?

1. Suppose you use technology to fit a straight line equation to some data and get the result y = 13.46582674645x + 0.003756193675.

a. Express your equation, with each parameter rounded off to 5 places after the decimal point (5 decimal places).
b. Express your equation, with each parameter rounded off to 5 significant digits (5 significant figures).
c. Express your equation, with each parameter truncated to 5 significant digits, using the ellipsis (...) notation.

Answers:

(a) y = 13.46583x + 0.00376
(b) y = 13.466x + 0.0037562
(c) y = 13.465...x + 0.0037561....

2. Suppose you use technology to fit a straight line equation to some data and get the result y = 0.015679835x + 200.0036987.

a. Express your equation, with each parameter rounded off to 5 places after the decimal point (5 decimal places).
b. Express your equation, with each parameter rounded off to 5 significant digits (5 significant figures).
c. Express your equation, with each parameter truncated to 5 significant digits, using the ellipsis (...) notation.

5. You help run a local coffeehouse, and have had the same musical group perform three years in a row. You have charged different admission prices each year, and attendance has varied. The first year you charged $7 per ticket and 100 people bought tickets. The second year 80 tickets were sold at $8 per ticket. Last year you charged $9 and sold 70 tickets. You have agreed to guarantee to pay the band $300 this year. We implicitly formulated a model for the number of tickets sold in terms of the price of each ticket:

Verbal Definition: n(p) = the number of tickets sold if they cost p dollars per ticket.
Symbol Definition: n(p) = -15p + 203.33..., for 0 is less than or equal to p, which is less than or equal to 13.50.
Assumptions: Certainty and divisibility. Certainty implies that the sales will follow the demand function exactly, no discounts, and all pay full price. Divisibility implies that fractions of tickets can be sold.

You eventually need to decide what price to charge for tickets. Suppose you believe that you will receive an additional $2 in revenue for food and drinks on average from every person attending the concert. Assume that the food and drink have been donated at no cost.

(a) Formulate and fully define a model for the revenue from food and drink in terms of p.
(b) Now find an expression for the total revenue (from ticket sales and food and drink) for the concert, in terms of p.
(c) Now fully divine a model for the profit (including both categories of revenue).

Answers:

(a) Verbal definition: r(p) = the revenue, in dollars, from food and drink when tickets are sold at p dollars per ticket.

Symbol definition: r(p) = 2 (-15p + 203.33...), for 0 is less than or equal to p, which is less than or equal to 13.50.

Assumptions: Certainty and divisibility. Certainty implies that the relationship is exact. Divisibility implies that fractions of tickets can be sold.

(b) R(p) = (p + 2) (-15p + 203.33...), for 0 is less than or equal to p, which is less than or equal to 13.50.

(c) Verbal Definition: P(p) = the total profit from ticket sales and food and drink when tickets are sold at p dollars per ticket.

Symbol definition: P(p) = (p + 2) (-15p + 203.33...) - 300, for 0 is less than or equal to p, which is less than or equal to 13.50.

Assumptions: Certainty and divisibility. Certainty implies that the relationship is exact. Divisibility assumes that fractions of tickets can be sold.

Note: (This is not the same question, difference is an additional $3 instead of $2):

6. You help run a local coffeehouse, and have had the same musical group perform three years in a row. You have charged different admission prices each year, and attendance has varied. The first year you charged $7 per ticket and 100 people bought tickets. The second year 80 tickets were sold at $8 per ticket. Last year you charged $9 and sold 70 tickets. You have agreed to guarantee to pay the band $300 this year. We implicitly formulated a model for the number of tickets sold in terms of the price of each ticket:

Verbal Definition: n(p) = the number of tickets sold if they cost p dollars per ticket.
Symbol Definition: n(p) = -15p + 203.33..., for 0 is less than or equal to p, which is less than or equal to 13.50.
Assumptions: Certainty and divisibility. Certainty implies that the sales will follow the demand function exactly, no discounts, and all pay full price. Divisibility implies that fractions of tickets can be sold.

You eventually need to decide what price to charge for tickets. Suppose you believe that you will receive an additional $3 in revenue for food and drinks on average from every person attending the concert. Assume that the food and drink have been donated at no cost.

(d) Formulate and fully define a model for the revenue from food and drink in terms of p.
(e) Now find an expression for the total revenue (from ticket sales and food and drink) for the concert, in terms of p.
(f) Now fully divine a model for the profit (including both categories of revenue).

3. Could the following data set be reasonably modeled for a linear function?

0 1 2 3 4 5 6 7
1.5 2.4 3.6 4.5 5.6 6.7 7.4 8.2

Answer: This data could be reasonably modeled by a linear function. (Explain why).

15. You have been training to run in your first marathon. You certainly don't expect to win, but you would like to make a decent showing, hopefully finishing in 4 hours. The race is usually 26 miles, 385 yards in length. (There are 1760 yards in a mile.) Your strategy is to set your pace and stick with it throughout the race.

(a) Determine what this pace must be in minutes per mile if you are to finish in 4 hours.
(b) Write the distance you should have covered as a function of the time elapsed.
(c) Sketch a graph of this function.

Answer for (a): Your pace should be about 9.15 minutes per mile to finish in 4 hours.

16. On a regular class day, you start out from your room for class, which starts in 10 minutes. The classroom is approximately a quarter of a mile away. You start out at a leisurely pace and arrive at class early. Suddenly you remember that you left your assignment in your room. You double your speed, go back to your room, grab the paper and go to class, arriving just in time.

(a) Sketch a graph of the relationship between time and distance from your room, with time on the horizontal axis.
(b) Is the distance a function of the time? Is it linear?
(c) Sketch a graph of the relationship between time and distance, with distance on the horizontal axis.
(d) Is the time a function of the distance? Is it linear?

3. Given the following data set, what types of functions would you consider for a model?

1 2 3 4 5 6 7 8 9 10
119 110 98 80 68 45 37 25 12 3

Answer: Linear (Explain why).

3. Given the following data set, what type of model would you fit for interpolation? Explain your reasons for choosing that type of model.

0 1 2 3 4 5 6 7 8 9 10
1.01 1.02 1.17 1.28 1.23 1.18 1.17 1.15 1.21 1.29 1.37

Answer: Cubic or quartic.

7. Everyone should be concerned with our growing debt in the United States. The following table shows the per capita public debt of the United States (in dollars):

Fiscal Year Per Capita ($)
1975 2475
1976 2852
1977 3170
1978 3463
1979 3669
1980 3985
1981 4338
1982 4913
1983 5870
1984 6640
1985 7598
1986 8774
1987 9615
1988 10534
1989 11545
1990 13000
1991 14436
1992 15846
1993 16871
1994 18026

(a) Fit a model to the data.
(b) Explain why you chose this model.
(c) What does your model predict for the per capita debt in 1995? 1996?
(d) Do you think these are reasonable predictions? Explain your answer.
(e) If possible, locate the actual figures for 1995 and 1996. Are the predictions from your model close to the actual figures?

Answers:

7. (a) Answers may vary. One possible model is this:

Verbal Definition: f(x) = the per capita public debt of the United States, in dollars, x years after mid-1975. x 0.10903...x
Symbol Definition: f(x) = 2461.3...(1.1152...) or 2461.3...e , for 0 which is less than or equal to x, which is less than or equal to 19.
Assumptions: Certainty and divisibility. Certainty implies that the relationship is exact. Divisibility implies that any fractional values of dollars and years are possible.

(b) Answers may vary.

(c) Answers may vary.
mid-1995: f(20) = 21, 796 ($21, 796)
mid-1996: f(21) = 24,307 ($24,307)

(d) Answers may vary. The predictions to seem to be very high, but public debt has been growing sharply.

(e) Answers may vary as data becomes available.

8. Show below is the percent of federal outlays for the interest paid on public debt in the United States:

Fiscal Year % of Federal Outlays
1975 9.8
1976 10.0
1977 10.2
1978 10.6
1979 11.9
1980 12.7
1981 14.1
1982 15.7
1983 15.9
1984 18.1
1985 18.9
1986 19.2
1987 19.5
1988 20.1
1989 21.0
1990 21.1
1991 21.5
1992 21.1
1993 20.8
1994 20.3

(a) Fit a model to the data.
(b) Explain why you chose this model.
(c) What does your model predict for the percent expenditures in 1995? 1996?
(d) Do you think these are reasonable predictions? Explain your answer.
(e) If possible, locate the actual figures for 1995 and 1996. Are the predictions from your model close to the actual figures?
(f) Does this model present a different picture from that given in the previous model?

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A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important .

Hi, I am having trouble doing these problems listed below. Please show me how to solve these problems for future reference. Thank you very much.

I would like for you to show me all of your work/calculations and the correct answer to each problem.

1. Given the following table:

x 2 4 8 9 10 11 10 15
y 4 5 7 9 11 13 15 17

(a) Is y a function of x? Explain your answer.
(b) Is x a function of y? Explain your answer.

Answers:

1. (a) No, y is not a function of x because there is not exactly one y value for each x value.
(b) Yes, x is a function of y because there is exactly one x value for each y value.

2. Given the following table:

x 1.22 1.43 1.08 1.99 1.51 1.13 1.51 1.72
y 4 5 7 9 10 11 9 15

(a) Is y a function of x? Explain your answer.
(b) Is b a function of y? Explain your answer.

Answers:

2. (a) yes, y is a function of x because there is exactly one y value for each x value.
(b) No, x is a not a function of y because there is not exactly one x value for each y value.

13. A basketball team keeps a record of the number of points scored by the star player in each game:

Game 1  23 points
Game 2  18 points
Game 3  21 points
Game 4  18 points
Game 5  19 points
Game 6  22 points
Game 7  17 points
Game 8  24 points

(a) Is the number of points scored a function of the game number? Why or why not?
(b) Is the game number a function of the number of points scored? Why or why not?

Answers:

(a) Yes. The number of points is a function of the game because there is exactly one point value for each game.
(b) No. The game number is not a function of the points scored because the value of 18 points could come from Game 2 or 4.

14. The table below gives the results of a survey showing the price charged for a sweatshirt and the number of students saying that the price is the most they would pay, to the nearest $2.50.

Price ($) 0 2.50 5.00 7.50 10 12.50 15 17.50 20 22.50 25

# of Students 9 3 2 12 15 18 17 12 8 3 1

(a) Is the number of students a function of the price? Why or why not?
(b) Is the price a function of the number of students? Why or why not?

1. Suppose you use technology to fit a straight line equation to some data and get the result y = 13.46582674645x + 0.003756193675.

a. Express your equation, with each parameter rounded off to 5 places after the decimal point (5 decimal places).
b. Express your equation, with each parameter rounded off to 5 significant digits (5 significant figures).
c. Express your equation, with each parameter truncated to 5 significant digits, using the ellipsis (...) notation.

Answers:

(a) y = 13.46583x + 0.00376
(b) y = 13.466x + 0.0037562
(c) y = 13.465...x + 0.0037561....

2. Suppose you use technology to fit a straight line equation to some data and get the result y = 0.015679835x + 200.0036987.

a. Express your equation, with each parameter rounded off to 5 places after the decimal point (5 decimal places).
b. Express your equation, with each parameter rounded off to 5 significant digits (5 significant figures).
c. Express your equation, with each parameter truncated to 5 significant digits, using the ellipsis (...) notation.

Answers:

(a) y = 0.01568x + 200.00370
(b) y = 0.015680x + 200.00
(c) y = ...

Solution Summary

The expert examines business calculus functions.

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