Name
Student ID
MATH 1040 – Business Mathematics
Fall 2024 – Midterm Examination
ˆ Calculators and one A4 sheet (both sides) of written notes are permitted
ˆ Time allowed: 120 minutes
ˆ Number of questions: 6
ˆ Number of pages, including exam cover sheet: 8
ˆ You must write your answer in pen. Paper written in pencil that are illegible or difficult
to read will be deemed incorrect and will receive a zero mark
ˆ Instructors’ name: Nguyen Ngoc Phan
Total points: 40
College of Business and Management, Vin University, Hanoi, Vietnam
1. (5 points) Company A needs $3,000,000 in 6 years to replace a machine. The company
sets up a sinking fund for that purpose. The company’s deposits earn 5.2% interest,
compounded quarterly.
(a) Find the payment it should make at the end of each quarter into the sinking fund.
(2 points)
(b) Find the present value of this annuity.
(2 points)
(c) Discuss the advantages of annuity over the lumpsum deposit in this particular
case.
(1 point)
2. (10 points) A company has an empty department in one of its factories that could be
used to expand the production of current products or manufacture new goods. Two
proposals relating to the use of this space have been submitted. Only one of them
can be accepted, since each project would fully utilize the department. The following
information was obtained for the proposals:
Year
Proposal A annual cash flow
Proposal B annual cash flow
0
-$25 million
-$35 million
1
$16 million
$10 million
2
$12 million
$15 million
3
$10 million
$20 million
4
$8 million
$15 million
5
$6 million
$10 million
(a) Calculate the payback period, accounting rate of return, internal rate of return
and net present value for each of the two projects. Given that the company’s cost
of capital is 10%.
(8 points)
(b) Which project would you recommend the company to choose? Justify your answer.
(2 points)
3. (4 points) Find the producers’ surplus at a price level of p̄ = $55 for the price–supply
equation
p = S(x) = 15 + 0.1x + 0.003x2
4. (5 points) The price–demand equation for hamburgers at a fast-food restaurant is
x + 400p = 3, 000.
(a) Find the elasticity of demand at the current price of $3.50.
(4 points)
(b) If the price is increased by 5%, will revenue increase or decrease?
(1 point)
5. (6 points) An oil spill has fouled 200 miles of Pacific shoreline. The oil company
responsible has been given 14 days to clean up the shoreline, after which a fine will
be levied in the amount of $10,000/day. The local cleanup crew can scrub five miles
of beach per week at a cost of $500/day. Additional crews, which can also scrub five
miles of beach per week, can be brought in at a cost of $18,000 plus $800/day for each
crew. How many additional crews should be brought in to minimize the total cost to
the company?
6. (10 points) The rate at which medical samples arrive at a laboratory t hours after 8.00
am on a particular day can be modeled using the function
m(t) = 400 t3/2 e−0.8t
for 0 ≤ t ≤ 9
where the rate at which medical samples arrive at the laboratory, m(t), is measured in
samples per hour.
(a)
i. Show that m′ (t) = 600 t0.5 e−0.8t − 320 t1.5 e−0.8t .
(2 points)
ii. Hence, using an algebraic approach, show that the rate at which medical
samples arrive at the laboratory is maximized at t = 15/8, according to the
model.
(2 points)
When a medical sample arrives at the laboratory, it is placed in a queue before
being processed. The medical samples are processed at a constant rate of 140
samples per hour. Let the function p(t) = 140 represent the rate at which medical
samples are processed.
The figure below shows the graphs of y = m(t) and y = p(t), where 0 ≤ t ≤ 9.
The graphs of y = m(t) and y = p(t) intersect at t = 0.735 and t = 3.83.
(b) At 8.00 am (t=0), the laboratory had 500 medical tests in the queue to be processed.
i. Explain why the number of items in the queue is decreasing until 0.735 hours
after 8.00 am.
(2 points)
ii. During the time that the number of items in the queue was increasing, the
queue increased by K medical samples. Determine the value of K, correct to
the nearest integer.
(2 points)
iii. Determine the number of medical samples in the queue at 5.00 pm.(2 points)