Name:
Math 140
Takehome Exam
Answer the questions in the spaces provided on the question sheets. If you
run out of room for an answer, please feel free to add more pages. Most
questions do not necessarily have just one right answer. There WILL be only
one correct equation, but there is more than one correct conclusion based on
the equation. Thus you need to justify any conclusion that you reach by
using an equation AND an explanation. Clearly show all work for full
credit.(100 points total)
(20pts.) 1. Crime scene investigators can estimate the time of death of a deceased individual by
taking the temperature of the liver due to the fact that the human liver obeys Newton’s
Law of Cooling. Given that the typical temperature of a live human body is 98.5◦ F and
that the liver cools at a rate of 1.5◦ F per hour when the ambient temperature is 65◦ F .
About how long has a body exposed to 25◦ F been deceased if the liver temperature is
45◦ F when the investigators arrive on scene. (NOTE: This is actually how they estimate
time of death.)
(20 pts.) 2. The volume V of an ideal gas varies directly with the temperature T and inversely with
the pressure P . Air is considered to be an ideal gas. Your company produces a 25 gallon
air compressor (about 95.5 Liters) that cuts off when the pressure in the tank reaches 100
psi (about 6.8 atmospheres), with a motor that can put out 13.1 cubic feet per minute
of compressed air, and assumes that the maximum operating temperature of the tank is
90◦ F (about 305.4◦ K).
(a) Write the variation equation in terms of P , V , and T that models the above problem.
(b) What is the constant of variation k in terms of metric units?
(c) The tank is designed to safely hold 8 atmospheres of airpressure but the company
sets the cutoff at 6.8 atmospheres since the actual safe maximum air presuure can
vary by 10% from tank to tank due to the quality of the welding on the tank. A
frequent customer calls and states that they are interested in purchasing the compressor, but their shop reaches 120◦ F (about 322.1◦ K) frequently. He states that
this product meets his needs, but he is concerned since his shop reaches temperatures well above the maximum operating temperature of the compressor. What
would you recommend? (This compressor or another one that your company offers.)
Page 2
(20 pts.) 3. You work in the financial department for a small buisness. The owner comes up to you
and says that the company is looking at moving its account to a different bank. She
informs you that she is considering three options:
1. Account 1 has a 0% monthly interest rate, charges a monthly service fee of $12.00,
and charges $1.25 for each wire transfer.
2. Account 2 has a 0.1% monthly interest rate, charges a monthly service fee of $15.00,
allows 15 free wire transfers a month, and charges $1.25 for each wire transfer after.
3. Account 3 has a 0.7% monthly interest rate, charges a monthly service fee of $250.00,
but all wire transfers are free.
(a) Given that your company maintains an average balance of $80, 000.00 a month and
performs two direct deposits (which are wire transfers) for each of its 50 employees
per month, create an equation, in terms of the number of employees, for each of the
above options that models the cost associated with each account per month.
(b) Which account option would you recommend?
(c) The owner informs you that the company put in a bid for a major contract that
will require the company to hire an additional ten employees and will cause average
balance of the company account to increase to $95, 000.00 a month. Which option
would you recommend under this scenario.
Page 3
(20 pts.) 4. An advertising company approaches you and states that if you are willing to run advertisements with them they can model the number of units sold by:
U (d) =
800
1 + 12e−0.027d
where U is the amount of units sold in thousands per year and d is the amount you
spend on advertising in thousands of dollars.
(a) Given that you currently sell 70, 000 units without any advertising and that the
approximate population of your market is 700, 000 people, do you think their model
is accurate? Why?(Hint: I would say that it isn’t.)
(b) You decide to perform your own research and come up with the model:
U (d) =
700
1 + 9e−0.01d
where U is the amount of units sold in thousands per year and d is the amount
you spend on advertising in thousands of dollars. Given that you make a profit of
one dollar per unit sold, do you think that you should start advertising? (Hint:
Consider, where is the function increasing most rapidly.)
(c) You call your director of manufactoring and discover that you can lower your cost
by $0.50 per unit if your produce over 150,000 units. Do you think you should start
advertising with this added piece of information?
Page 4
(20 pts.) 5. Iodine 131 is a radioactive material that decays according to the function
A(t) = Ao e−0.087t
where Ao is the intial amount present and A is the amount present at time t (in days).
Assuming a sample of 100 grams:
(a) What is the decay rate of iodine 131?
(b) How much iodine 131 is left after 9 days?
(c) When will 70 grams of iodine 131 be left?
(d) What is the half-life of iodine 131?
(e) After the release of radioactive material into the atmoshpere from a nuclear power
plant at Chernobyl in 1986, the hay in Austria was contaminated by iodine 131. If
it is safe to feed the hay to cows when 10% of the iodine 131 remains, how long did
one need to wait before the hay was used.
Page 5