Granillo, Yvette – Homework 12 – Due: Nov 17 2005, 3:00 am – Inst: Edward Odell
1
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printout
should
have
19
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
The due time is Central
time.
001
(part 1 of 1) 10 points
A trailer rental agency rents 10 trailers per
day at a rate of $8 per day. It discovers that
for each $2 increase in rate, one fewer trailer
is rented.
Determine the rate,
r
max
, which
maximizes the rental income.
1.
r
max
= $14
correct
2.
r
max
= $16
3.
none of these
4.
r
max
= $18
5.
r
max
= $12
Explanation:
Let
r
be the daily rate of renting a trailer.
Then the number of trailers rented daily is
10

(
r

8)
2
.
Hence the daily income,
I
(
r
), of the agency is
given by
I
(
r
) =
r
10

(
r

8)
2
¶
.
The income will be maximized, therefore, at
the critical points of
I
(
r
). Now after differen
tiation,
I
0
(
r
) =
10

(
r

8)
2
¶

r
2
.
Thus the critical points of
I
(
r
) occur at
10

(
r

8)
2
¶

r
2
= 0
,
i.e., r
= 14
.
Consequently,
r
max
= $14
is the rate that maximizes income.
keywords: Stewart5e,
002
(part 1 of 1) 10 points
Circuit City has been selling 100 television
sets a week at $600 each.
A market survey
indicates that for each $20 rebate offered to
a buyer, the number of sets sold will increase
by 5 per week.
How large a rebate should Circuit City offer
a buyer in order to maximize its revenue?
1.
rebate = $115
2.
rebate = $100
correct
3.
rebate = $95
4.
rebate = $110
5.
rebate = $105
6.
none of these
Explanation:
Let $20
x
be the rebate offered to a buyer.
Then the price of a TV will be $(600

20
x
)
and the number of sets sold at this price will
be 100 + 5
x
. The revenue with this rebate is
thus
R
(
x
) = (600

20
x
)(100 + 5
x
)
= 100(30

x
)(20 +
x
)
= 100(600 + 10
x

x
2
)
.
But then
R
0
(
x
) = 100(10

2
x
)
,
while
R
00
(
x
) =

100
×
2
<
0
.
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Granillo, Yvette – Homework 12 – Due: Nov 17 2005, 3:00 am – Inst: Edward Odell
2
Consequently, the Revenue is maximized at
x
0
= 5, in which case the
rebate = $100
.
keywords: Stewart5e,
003
(part 1 of 1) 10 points
A high speed office copier has an initial
price of $3200.
A service contract cost $260
for the first year and increases by $100 per
year thereafter. It can be shown that over
x
years the total cost of the copier is given by
C
(
x
) = 3200 + 210
x
+ 50
x
2
.
When is the average cost per year smallest?
(This is often referred to as the
replacement
time
for a piece of equipment.)
1.
replacement time = 9 years
2.
replacement time = 7 years
3.
replacement time = 8 years
correct
4.
replacement time = 11 years
5.
replacement time = 10 years
Explanation:
The average cost
c
(
x
) =
C
(
x
)
x
=
3200 + 210
x
+ 50
x
2
x
=
3200
x
+ 210 + 50
x .
In practical terms it represents the total cost
(initial cost plus maintenance costs) of the
computer averaged out over an
x
year period.
The marginal average cost, therefore, is given
by
c
0
(
t
) =

3200
x
2
+ 50
,
and so the only critical point of
c
(
x
) for
x >
0
is the positive solution of
c
0
(
x
) = 0,
i.e.
,
replacement time =
r
3200
50
= 8 years
.
keywords: Stewart5e,
004
(part 1 of 2) 10 points
The marketing department of a company
has determined that the cost (in dollars) of
producing
x
units of a new product is given
by
C
(
x
) = 300 + 180
x,
while the demand equation for that product
will be given by
p
= 300

3
100
x.
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 Spring '08
 schultz
 Inverse function, Injective function, Edward Odell, Granillo

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