Rex Carr, who owns a junk yard, can use one of two methods to destroy cars. The

Question

Rex Carr, who owns a junk yard, can use one of two methods to destroy cars. The first involves purchasing a hydraulic car smasher that costs 3200 a year to own and then spending $1 for every car smashed into oblivion; the second method involves purchasing a shovel that will last one year and costs $10 and paying his brother to bury the cars at a cost of $5 each. Write down the total cost functions for the two methods, where y is output per year: C_1(y) = C_2(y) = Compute the average and marginal costs of each method: If Rex wrecks 40 cars per year, which method should he use? If Rex wrecks 50 cars per year, which method should he use? What is the smallest number of cars per year for which it would pay him to buy the hydraulic smasher?

Explanation / Answer

(a) i. C1(y) = $ 200 + $1(y) = 200 + y

y is the quantity of cars that are smashed.

Therefore, C1(y) = 200 + y

ii. C2(y) = $ 10 + $5(y) = 10 + 5y

Therefore, C2(y) = 10 + 5y

(b) i. AC1(y) = TC1(y)/y or C1(y)/y = (200 + y)/y = 200/y + 1

MC1(y) = derivative of C1(y) with respect to y = 1

ii. AC2(y) = TC2(y)/y or C2(y)/y = (10 + 5y)/y = 10/y + 5

MC1(y) = derivative of C2(y) with respect to y = 5

(c) If y = 40

C1(40) = 200 + 40 = 240

C2(40) = 10 + 5(40) = 10 + 200 = 210

It shows that second method should be used because it costs less.

(d) when y = 50

C1(50) = 200 + 50 = 250

C2(50) = 10 + 5(50) = 10 + 250 = 260

It shows that first method should be used because it costs less.

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