Homework
# 5
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Homework # 5
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Due Date March 12
Find the indicated integral.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1. A manufacturer estimates that the marginal cost of producing q units of a certain
commodity is C(q) = 3q2 +24q +48 dollars per unit. If the cost of producing 10 units is
$5,000, what is the cost of producing 30 units?
2. After initiating an advertising campaign in an urban area, a satellite dish provider
estimates that the number of new subscribers will grow at a rate given by N(t) =154t2+37
subscribers per month where t is the number of months after the advertising begins. How
many new subscribers should be expected 8 months from now?
3. A manufacturer estimates marginal revenue to be R_(q) _ 100q_1/2 dollars per unit when
the level of production is q units. The corresponding marginal cost has been found to be
0.4q dollars per unit. Suppose the manufacturer’s profit is $520 when the level of
production is 16 units. What is the manufacturer’s profit when the level of production is
25 units
4. In following exercises, the slope f `(x) at each point (x, y) on a curve y =f (x) is given
along with a particular point (a, b) on the curve. Use this information to find f (x).
5. Bob is taking a learning test in which the time he takes to memorize items from a given
list is recorded. Let M(t) be the number of items he can memorize in t minutes. His
learning rate is found to be M(t) = 0.4t =0.005t2
a. How many items can Bob memorize during the first 10 minutes?
b. How many additional items can he memorize during the next 10 minutes (from time t
=10 to t= 20)?
6. At a certain factory, when K thousand dollars is invested in the plant, the production Q is
changing at a rate given by Q(K) =200K-2/3 units per thousand dollars invested. When
$8,000 is invested, the level of production is 5,500 units.’
a. Find a formula for the level of production Q to be expected when K thousand dollars is
invested.
b. How many units will be produced when $27,000
is invested?
c. What capital investment K is required to produce 7,000 units?
7. Suppose it has been determined that the marginal revenue associated with the production
of x units of a particular commodity is R`(x) =240 -4x dollars per unit. What is the
revenue function R(x)? You may assume R(0) = 0. What price will be paid for each unit
when the level of production is x =5 units?
8. The marginal revenue from the sale of x units of a particular commodity is estimated to
be ?′ = 50 + 3.5??−0.01?
2
dollars per unit where R(x) is revenue in dollars.
a. Find R(x), assuming that R(0) = 0
b. What revenue should be expected from the sale of 1,000 units?
9. The concentration C(t) in milligrams per cubic centimeter (mg/cm3) of a drug in a
patient’s bloodstream is 0.5 mg/cm3 immediately after an injection and t minutes later is
decreasing at the rate f?′(?) =
−0.01?0.01?
(?0.01?+1)2
A new injection is given when the concentration drops below 0.05 mg/cm3.
a. Find an expression for C(t).
b. What is the concentration after 1 hour? Ater 3 hours
10. The owner of a fast-food chain determines that if x thousand units of a new meal item are
supplied, then the marginal price at that level of supply is given by
?(?) =
?
(? + 3)2
where p(x) is the price (in dollars) per unit at which all x meal units will be sold. Currently, 5,000
units are being supplied at a price of $2.20 per unit.
a. Find the supply (price) function p(x).
b. If 10,000 meal units are supplied to restaurants in the chain, what unit price should be charged
so that all the units will be sold?
