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Harvard Mathematics For Management Final Exam 2023

Q1. A housepainter buys 40 gallon-size cans of paint for $920. Some of these are blue paint that cost $30 per gallon and the rest are white paint that cost $20 per gallon. How many gallons of white paint does the painter buy?

Q2. What is the value of x if log₂ x = 3

Q3. What is the slope of the tangent line to the curve y = x³ – 4x at x = -3?

Q4. Information about a stock is given in the table below

Harvard Mathematics For Management Final Exam

Using the file data_stock_option.xlsx, what is the correct price for a 6-month European put option?

$0.44
$0.85
$3.73
$4.44

Q5. The minimum value of the function f(x) = x² + 2x + 3 occurs at which value of x?

-1
0
1
2

Q6. You determine the following two variables for the next 10 people who come into the classroom

I. The person’s shoe size.
II. The length of the person’s foot.

Which of these two measurements is a continuous random variable?

I only
II only
Both I and II
Neither I nor II

Q7. What is the product of the roots of 2x² + 9x – 5 = 0 ?

5
-5
5/2
-5/2

Q8. Suppose that your bank pays you 15% annual interest that is compounded quarterly. What is the effective annual interest rate (rounded to the nearest hundredth of a percent)?

15.00%
15.56%
15.87%
16.18%

Q9. What are all the values of x for which y = x + x^-1 is convex?

Q10. The daily revenue figures for Acme Multiplex Cinema, for the past month, are in the file data_acme_multiplex.xlsx. What is the mean daily revenue, to the nearest whole number?

$814
$1,235
$24,080
$38,025

Q11. A gas station has monthly expenses, in dollars, of 5000 +2.2x, where x is the number of gallons of gas sold. It sells the gas at $2.25 per gallon. How many gallons do they need to sell to make a profit of $3000 per month?

16,000
40,000
100,000
160,000

Q12. A line has slope 3 and y-intercept 5. This line is the graph of which equation?

3x + y = 5
3x – y = 5
3x + y = -5
3x – y = -5

Q13. Assume that you are interested in buying a coupon bond. It has a maturity of 12 years, a coupon rate of 9%, and a par value of $100. The required annual return for this bond is 7%. Assume that interest is compounded annually. What is the fair price of this bond?

$85.68
$100.00
$114.05
$115.89

Q14. Suppose that your bank pays you a 15% annual interest rate, compounded quarterly, on your investments. If you deposit $200 today, how much money will you have in exactly four years?

$231.73
$320.00
$349.80
$360.45

Q15. What is the value of

Q16. Let f (x,y) = 5y + 10x What is the value of f (6,2)?

Q17. You get on an elevator on the first floor of a building with 10 floors. You push the button for the fifth floor. A random person gets on before the door closes and chooses a floor. What is the probability that the other person gets off the elevator before you? (Assume you exit together if you go to the same floor.)

3/10
1/3
2/5
1/2

Q18. What are all values of x that satisfy 4x – 1 < 6x + 9 ?

x > -4
x < -4
x > -5
x < -5

Q19. The Crimson Sox are playing a home game and have a 60% chance of winning. At the same time, the Northerners are playing a home game and have a 60% chance of winning. What is the probability that the Crimson Sox win and the Northerners lose?

Q20. The commute times (in minutes) for 10 employees of Red Banjo Brewing are 15, 58, 59, 60, 59, 54, 56, 59, 59, and 56. What is the mode, to the nearest minute, for this data set?

Q21. Two dice, one red and one blue, are thrown. What is the probability that the number on the blue die is 1 more than the number on the red die?

5/36
1/6
5/18
1/3

Q22. Let q represent the number of widgets sold each week by a store, p represent the price in dollars of a widget, and D represent the demand. Suppose that the demand function is q = D (p) = 120 / √p

What is the decrease in the demand when p is changed from $9 to $25?

Q23. A class has 30 students. The students answer a poll about whether they like cats or dogs. The total number of students in each category is given in the table below.

Harvard Mathematics For Management Final Exam

What is the probability that a randomly chosen student in this class likes cats, given that they like dogs?

3/10
9/20
1/2
3/5

Q24. Last month, the average weekday sale of Red Banjo IPA beer was 160 pints, with a standard deviation of 5 pints. Suppose that the data set is roughly symmetric and that last month’s sales were typical. What proportion of the time would you expect a weekday’s sales to be between 150 and 170 pints?

50%
68%
95%
99.7%

Q25. The annual revenue growth rates for five years for Stephen’s Suitcases are shown in the following table.

What is the Compound Annual Growth Rate (CAGR) for Stephen’s Suitcases?

-7.40%
-1.52%
1.52%
7.40%

Q26. Let p = the price in dollars charged for a washing machine by an appliance store. Suppose that the annual demand at that price will be 3200-5p and the annual supply will be 3p-400 . At what price will the annual supply equal the annual demand?

$133
$350
$450
$640

Q27. The one-way mileage numbers for the commute of seven FlatCola Cybersystems employees are: 8, 15, 25, 29, 23, 24, and 25. For this data set, which of the following inequalities is true?

Mode < Median < Mean
Mean < Median < Mode
Median < Mean < Mode
Median < Mode < Mean

Q28. Floormart had $500,000 in revenue in 2019. They anticipate revenue growth of 10% each year. If they achieve this goal, what will their revenue be in 2021?

$550,000
$600,000
$605,000
$666,000

Q29. Sally owns a restaurant in Wooville. The city is trying to get a minor league team to move there in 2021, either to location A (near her restaurant) or location B. The team might stay in location C in another city. The probability of these events, and her estimated annual profit in 2021 are shown in the table below.
Harvard Mathematics For Management Final Exam

What is the standard deviation of the restaurant’s profit in 2021 (rounded to the nearest dollar)?

$61,612
$65,422
$71,181
$74,503

Q30. Today is January 1, 2020. On the first day of the years 2021 to 2028, you will receive $12,000. If these cash flows are discounted at 11% per year, what is the present value of these cash flows (rounded to the nearest dollar)?

$56,546
$61,753
$66,445
$84,000

Q31. Let q = the annual demand in thousands of tubes for Spiffy Whitening toothpaste and p = the price, in dollars, per tube for the toothpaste. Suppose that the demand curve is q = 400/ p² Which equation shows the inverse demand curve?

p = 20 / √q
p = 200 / √q
p = 20 / q
p = 20 √q

Q32. Rubberneck Tires produces a tire that will last for 70,000 miles on average. The mileage is a normal random variable with a standard deviation of 8,000 miles. They warranty the tire for 55,000 miles. What is the probability (rounded to the nearest hundredth) that a tire will fail to meet the warranty?

0.030
0.040
0.114
0.131

Q33. Choose the correct option to complete the sentence. Today is January 1, 2020. On the first day of the years 2021, 2022, 2023, and 2024, you will invest $12,000. If your expected rate of return is 7% per year, you will have ___________ on January 1, 2025 (rounded to the nearest dollar).

$48,000
$53,279
$57,009
$60,999

Q34. Which equation represents a power function that is decreasing and has a graph that passes through the point (2,1)?

y = 1/2 x^-1
y = 2 x^-1
y = 1/4 x²
y = 4 . 2^-x

Q35. Susan owns a stock and she is concerned that the price of her stock will go down. To hedge her position, she could buy a put option on the stock. True / False

Q36. If the following cash flows are discounted at 13%, what is the net present value (NPV) of this series of cash flows?

Harvard Mathematics For Management Final Exam

-$13.58
-$3.68
$3.68
$13.58

Q37. This coming year, Stephen’s Suitcases expects $50,000 in profit. Assume that the firm’s profits will grow at 4% per year forever and that cash flows are discounted at 7% per year. If profits are received at the end of each year, what is the present value of all the firm’s future profits?

$350,000
$714,286
$1,250,000
$1,666,667

Q38. Sally owns a restaurant in Wooville. The city is trying to get a minor league team to move there in 2021, either to location A (near her restaurant) or location B. The team might stay in location C in another city. The probability of these events, and her estimated annual profit in 2021 are shown in the table below.

Harvard Mathematics For Management Final Exam

What is the expected value of the restaurant’s profit in 2021?

$138,000
$145,000
$160,000
$190,000

Q39. If I Had A Hammer Hardware has sales of $5000 in its first month. Suppose sales were to increase by 4% every month. In how many months after the first month would the sales reach $9000? Round your answer to the nearest month.

Q40. If f (x) = x⁴ – 5x³ , then what is f ” (x) when x = -1?

Q41. Goldilocks and Company manufactures wooden platform beds. The length of the wooden slats used to make a full-size bed must be between 74.8 inches and 75.2 inches to be declared “just right.” Suppose that the length of the wood follows a normal random variable with mean 75 inches and standard deviation of 0.15 inch. To the nearest one-tenth of one percent, what is the percentage of wooden slats that are just right?

75.0%
81.8%
87.5%
90.9%

Q42. Suppose that the average cost per unit when manufacturing x units of a product each day is described by the function

C(x) = x³ – 1200x + 6000. Manufacturing how many units would minimize the average cost per unit?

Q43. Assume that you are interested in buying a zero coupon bond. It has a maturity of 25 years, a price of $462, and a par value of $1000. Assume that interest is compounded annually. What is the yield on this bond?

1.85%
2.16%
3.14%
8.66%

Q44. The attached file, data_red_banjo_pints_sold.xlsx, lists the daily sales of IPA beer, in pints, for the first three years of operations of Red Banjo Brewing Company. Construct a histogram with lower bin limit of 0 and bin width 50 for the variable IPA beer (pints). Which of the following best describes the sales data?

symmetric
negatively skewed
positively skewed
flat

Q45. What is the internal rate of return (IRR) of the following series of cash flows?

Harvard Mathematics For Management Final Exam

1.55%
7.16%
10%
13%

Q46. The CEO of Stephen’s Suitcases is considering buying a new machine that costs $6 million today. During each of the next 4 years, the CEO expects the machine to generate the following cash inflows. What is the payback period for this machine?

Less than 1 year
Between 1 and 2 years
Between 2 and 3 years
Exactly 3 years

Q47. The asking prices of HomeXpress’s nine most recent listings are in the attached file, data_homexpress.xlsx. Which of the following is the best description of the data set, based on comparing the mean and the median?

symmetric
negatively skewed
positively skewed
bimodal

Q48. A manager at Red Banjo Brewing wonders if their customers drink more IPA beer when the weather gets warmer. The attached file, data_red_banjo_temperature.xlsx, contains two weeks’ worth of data on the daily high temperature, along with the number of pints of IPA beer sold that day. What is the correlation coefficient of high temperature and IPA beer?

0.96
0.72
0
-0.30

Q49. Suppose that your bank pays you a 12% annual interest rate, compounded continuously, on your investments. If you deposit $250 today, how much money will you have in exactly six years?

$430.00
$493.46
$513.61
$527.62

Q50. The selling prices of the homes currently represented by Ravenweyr Realty are in the attached file, data_ravenweyr.xlsx.

Which of the following choices best describes the set of selling prices of these homes?

symmetric
negatively skewed
positively skewed
bimodal

Q51. The owner of a bar knows that only 1/3 of his customers between 6 and 7 pm will order dinner. Three people independently enter the bar at 6 pm. What is the probability that at least one of these people will order dinner?

1/27
8/27
19/27
26/27

Q52. Consider the following two points on two curves:

I (0,0) on f (x) = x³

II (0,0) on g (x) = x⁴

Which of these are inflection points?

I only
II only
Both I and II
Neither I nor II

Q53. The PTA sells 100 tickets for a raffle and puts them in a bowl. They will randomly pull out a ticket for the first prize and then another ticket for the second prize. You have 10 tickets and your friend has 10 tickets. What is the probability that your friend wins the first prize and you win the second prize?

1/5
1/10
1/99
1/100

Q54. Red Banjo Brewing is previewing a potential new offering: Chili Chocolate Stout. The numbers of pints of Chili Chocolate Stout consumed each day last week were 3, 4, 2, 3, 1, 5, and 4. For this data set, what is the sample standard deviation, rounded to the nearest tenth?

Q55. The ratio of expenses for merchandise to cost of labor for a small company is 5:3. They spent a total of $15,000 for these two costs during one week. How much did they spend on labor?

$5000
$5625
$9000
$9375

Q56. What is the value of the expression 40 + 5 + 3 x 2³

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