Name: February 11, 2013

PRACTICE PROBLEMS FOR MIDTERM EXAM-ANSWERS 1) The total cost to produce 10 calculators is $100. The variable cost per calculator is $4. (a) Find the fixed cost 10 4 4 10 40 60

100

(b) What is the cost function?

4 60

2) An insurance company claims that its monthly revenue in dollars is given by R(x) = 125x and its monthly cost is: C(x) = 100x + 5000, where x represents the number of policies in thousands. Find the break-even point R(x) = C(x) 125x = 100x + 5000 25x = 5000 x = 200

3) The total annual enrollment ( in millions) in U.S elementary schools for the years 1975-1996 is given by the model: E ( x) 0.058 x 2 1.162 x 50.604 , where x = 0 corresponds to 1975, and x = 1 corresponds to 1976 and so on. For this period: (a) When was the enrollment the lowest? x

. . . .

10.01

10

The enrollment was the lowest in 1985.

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(b) What was the enrollment to the nearest tenth of a million? E (10) 0.058(10) 2 1.162(10) 50.604 E (10) 5.8 11.62 50.604 E (10) 5.8 11.62 50.604 E (10) 44.784 44.8

4) Solve for x:

x 5 5 x 3 7 14 4 5 x5 x 3 28 28 28 7 14 4 4 x 5 10 7 x 3

4 x 20 10 7 x 21 4 x 7 x 31 20 11x 11 x 1

5) A company sells x units of a product whose profit function is given by the function: P ( x) 30 x x 2 . (a) How many units should the business sell in order to maximize profits? 15

x

(b) How much is the maximum profit?

15 30 15 15 450 225 225

(c) What is the maximum number of units the business can sell to make a profit? First find when P(x) = 0 and then take one unit less 0 30 x x 2 0 x 30 x x 0 & x 30 The business can sell 29 units to make some profit, thus x = 29. 2

6) Find the quadratic function whose vertex lies at the point (-4, 2) and passes through the point (-2, 6) f ( x ) a ( x h) 2 k f ( x) a ( x (4)) 2 2 6 6 = a(-2 4) 2 2 = 4a+2 a=1

thus equation is: f ( x) ( x 4) 2 2

7) An electric company produces 42-inch plasma HDTV’s that sells to retailers for $550. Producing and selling this product involves a monthly fixed cost of $213,000, and the cost for producing each component is $250. Let x represent a 42-inch plasma TV. a) b) c) d) Write the cost function C(x) for this component Write the revenue function R(x) for this component Write the profit function P(x) for this component What profit is gained or lost if 500 TVs were sold? a) C ( x ) 250 x 213, 000 b) R ( x ) 550 x P( x) R( x) C ( x) c) P( x) 550 x 250 x 213, 000 P( x) 550 x 250 x 213, 000 P( x) 300 x 213, 000

P (500) 300(500) 213, 000

d) P (500) 150, 000 213, 000 P (500) 63, 000 A Loss

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8) A company manufactures its products at a cost of $5 per unit and sells them for $12 per unit. If the firm’s fixed cost is $14,000 per month. a) Determine the firm’s break-even point. C ( x) 5 x 14000 R ( x) 12 x R ( x) C ( x) 12 x 5 x 14000 12 x 5 x 14000 7 x 14, 000 x 2, 000

b) What is the loss sustained by the firm if only 1200 units are produced and sold each month? P (1200) R(1200) C (1200) P (1200) 12(1200) 5(1200) 14, 000 P (1200) (12 5)(1200) 14, 000 P (1200) 7(1200) 14, 000 P (1200) 8400 14, 000 P (1200) 5600

c) What is the profit if 2,500 units are produced and sold each month? P (2500) R(2500) C (2500) P (2500) 12(2500) 5(2500) 14, 000 P (2500) (12 5)(2500) 14, 000 P (2500) 7(2500) 14, 000 P (2500) 17500 14, 000 P (2500) 3500

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d) How many units should the firm produce in order to realize a minimum monthly profit of $7,000?

R( x) C ( x) 7000 12 x (5 x 14000) 7000

12 x 5 x 14000 7000 7...