# A real estate office handles an apartment complex with 60 units

August 31, 2017

Question
1.A real estate office handles an apartment complex with 60 units. When the rent is \$615 per month, all 60 units are occupied. When the rent is \$660, however, the average number of occupied units drops to 57. Assume that the relationship between the monthly rent p and the demand x is linear. (The term demand refers to the number of occupied units.)

(a) Write a linear equation expressing x in terms of p.

(b) Predict the number of occupied units when the rent is set at \$735.
units

(c) Predict the number of occupied units when the rent is set at \$795.
units

2.The base and height of the trusses for the roof of a house are b = 34 feet and h = 4 feet, respectively (see figure). (Round your answers to two decimal places.)
(a) Find the distance from the eaves to the peak of the roof.
ft

(b) The length of the house is l = 50 feet. Use the result of part (a) to find the number of square feet of roofing.
ft2

3.Find the value(s) of x such that the distance between the points is 5. (Enter your answers as a comma-separated list.)(1,−1), (x,2)

x=

4.The annual inventory cost for a manufacturer is given by C = 1,008,000/Q + 7.2Q
where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 340 to 341. (Round your answer to two decimal places.)
\$

Compare this with the instantaneous rate of change when Q = 340. (Round your answer to two decimal places.)
\$ per unit

5.An environmental study indicates that the average daily levelPof a certain pollutant in the air, in parts per million, can be modeled by the equationP(n) =0.10

wherenis the number of residents of the community, in thousands. Find the rate at which the level of pollutant is increasing when the population of the community is16,000. (Round your answer to three decimal places.)

ppm/thousand residents

6.Find the slope of the graph at the given point. (10 − x)y2 = x3

7.Find the slope of the graph of the function at the given point. (If an answer is undefined, enter UNDEFINED.)x3 + y3 = 14xy, (7, 7)

8.Letxrepresent the units of labor andythe capital invested in a manufacturing process. When189,148units are produced, the relationship between labor and capital can be modeled by100x0.75y0.25=189,148.Find the rate of change ofywith respect toxwhenx=4000andy=200.

At (4000,200), dx/dy =

9.Find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. (If an answer does not exist, enter DNE.)f(x) =x3−27x,[0,6]

absolute maximum (x,y) =

absolute minimum(x,y) =

10.Find the number of unitsxthat produces the minimum average cost per unitCin the given equation.C = 0.08×3 + 59×2 + 1495

x = units

11.A real estate office handles a60-unit apartment complex. When the rent is\$530per month, all units are occupied. For each\$40increase in rent, however, an average of one unit becomes vacant. Each occupied unit requires an average of\$50per month for service and repairs. What rent should be charged to obtain a maximum profit?

12.When a wholesaler sold a product at \$40 per unit, sales were 250 units per week. After a price increase of \$5, however, the average number of units sold dropped to 225 per week. Assuming that the demand function is linear, what price per unit will yield a maximum total revenue?

13.An offshore oil well is 1 mile off the coast. The oil refinery is 8 miles down the coast. Laying pipe in the ocean is twice as expensive as laying it on land. Find x, and consider how finding xhelps determine the most economical path for the pipe from the well to the oil refinery. (See the figure below.)
x =

14.Find the particular solution that satisfies the differential equation and the initial condition. f '(x) = 12x − 12×3; f(3) = 1

f(x)=

15.Find the profit function for the given marginal profit and initial condition.

Marginal Profit Initial Condition
dP
dx
= −30x + 270
P(5) = \$610

p(x)=

16.Find the area of the region.

f(x) = x2 + 6x + 9
g(x) = 6x + 25

17.Find the consumer and producer surpluses by using the demand and supply functions, wherepis the price (in dollars) andxis the number of units (in millions).

Demand Function Supply Function
p = 390 − x p = 140 + x

consumer surplus \$ millions
producer surplus
\$millions

18.Find the consumer and producer surpluses by using the demand and supply functions, wherepis the price (in dollars) andxis the number of units (in millions).

Demand Function Supply Function
p = 1300 − 21x p = 44x

consumer surplus \$ millions
producer surplus
\$millions

19.

Evaluate the definite integral.

4 (−5x + 6) dx
Integral
1

20. Find the change in costCfor the given marginal. Assume that the number of unitsxincreases by5from the specified value ofx. (Round your answer to two decimal places.)

dC/dx= 24000/x^2 x=18

\$=

21.Evaluate the definite integral.
0
(t1/7 − t2/7)dt

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