Three Problems from samplefinal2007.pdf
2007 25. An apartment complex has 250 units. When the monthly
rent for each unit is $330, all units are occupied. Experience
indicates that for each $14 per month increase in rent, 4 units
will become vacant. Each rented apartment costs the owner of the
complex $50 per month to maintain. What monthly rent should be
charges to maximize profit? A) $125.5 B) $251 C) $376.5
D)$627.5
Solution Formula for demand is q = 250 (all) - (4 per $14)*(increase in rent from $330).
q(p)
=250 - (4/14) * (p - 330),
= (2410/7)-(2/7)p
Profit(p)
= (p -50) * q ,
dProfit
dp
= (2510/7)-(4/7)*p
The critical number is located at located at p = 627.5.
N.B. To express q as a linear function of p, we are actually
using the point-slope form: Point (p = 330, q = 250), slope
-4/14.
26. A commuter's train carries 600 passengers each day from a
suburb to a city. It now costs $1 per person to ride the train. A
study shows that 50 additional people will ride the train for each
5 cent reduction in fare. What fare should be charged in order to
maximize total revenue? A) 78 cents B) 79 cents C) 80 cents D)
85 cents.
Solution: Formula for demand (riders) is q = 600 - (50 per .05)*(increase in fare from $1.00).
q(p)
=600 - (50/.05) * (p - 1.00),
= 1600-1000p
R(p)
= p * q ,
dR
dp
= 1600-2000p
The critical number is located at located at p = 0.80.
28. A Florida citrus grower estimates that if 30 orange trees are
planted, the average yield per tree will be 200 oranges. The
average yield will decrease by 2 oranges per tree for each
additional tree planted on the same acreage. How many trees should
the grower plant to maximize the total yield? A) 65 trees B) 15
trees C) 35 trees D) 60 trees
Solution: Let T = trees planted.
Formula for output per tree (yield) is Y = 200 - (2 per 1 tree)*(increase in trees from 30).
Y(T)
=200 - (2/1) * (T -30),
= 260 - 2T.
R(T)
= T * Y ,
= 260 T - 2 T2
dR
dT
= 260- 4T.
The critical number is located at T = 65.
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