A retired auto mechanic hopes to open a rustproofing shop. Customers would be local new-car dealers. Two locations are being considered, one in the center of the city and one on the outskirts. The central city location would involve fixed monthly costs of $7,000 and labor, materials, and transportation costs of $30 per car. The outside location would have fixed monthly costs of $4,700 and labor, materials, and transportation costs of $40 per car. The dealer price at either location will be $90 per car.
(A) Which location will yield the greatest profit if monthly demand is (i) 200 cars, and (ii) 300 cars?
(B) At what volume of output will the two sites yield the same monthly profit?

Explanation:

As we know that

Total profit = Q(R-v) - FC

Provided that

Central location

Fixed cost or FC = $7000

Variable cost or v = $30

Revenue or R = $90

So,

Total profit = Q($90 - $30) - $7000

At Outside location

Fixed cost or FC = $4,700

Variable cost or v = $40

Revenue or R = $90

So,

Total profit = Q($90 - $40) - $4,700

Now

Q = 200 cars

So, Center profit = 200 cars ($90 - $30) - $7,000

= $12,000 - $7,000

= $5,000

At outside

= 200 cars ($90 - $40) - $4,700

= $10,000 - $4,700

= $5,300

So,

At Q = 200 cars Outside location is getting greater profits

Now

At Q = 300 cars

Center profit = $300 ($90 - $30) - $7,000

= $18,000 - $7,000 = 11,000

At Outside

= 300 ($90 - $40) - $4,700

= $15,000 - $4,700

= $10,300

So,

At Q = 300 cars Center location is getting higher profits

Now

By Equaling the profit equations

Q($90 - $30) - $7,000 = Q($90 -$40) - $4,700

$60Q - $7,000 = 50Q - $4,700

10Q = $2,300

Q = 230

So,

The two sites generate the same monthly benefit at 230 cars volume production

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