Question

An automobile dealer can sell 12 cars per day at a price of $13,000. He estimates that for each $300 price reduction he can sell two more cars per day. If each car costs him $10,000, and fixed costs are $1000, what price should he charge to maximize his profit? [Hint: Let x = the number of $300 price reductions.]

Answer 1

Answer:

$ 1100

Step-by-step explanation:

Let p(x) be the price he charges for a car after x price reductions.

Since car costs him $10,000, and fixed costs are $1000

p(x) = 13000-10000-1000-300x = 2000 - 300x

Let  q(x) = 12 + 2x the quantity of cars sold after x price reductions.

and the profit he can make are given by:

Profit = R(x) = (12+2x)(13,000 - 10,000 - 1,000 - 300x) dollars.

As you can see, maximizing profit , we have to check where P'(x) = 0

And P''(x)<0.

So we get x = 3.

Replacing the value we get we get 2000 - 300 X 3 = 2000 - 900

                                                                  = $ 1100

Answer 2

He should charge $12,400 per car to maximize his profit.

Given that an automobile dealer can sell 12 cars per day at a price of $ 13,000, and he estimates that for each $ 300 price reduction he can sell two more cars per day, to determine, if each car costs him $ 10,000, and fixed costs are $ 1000, what price should he charge to maximize his profit, the following calculation should be performed:

Sales - (Fixed costs + variable costs per car sold) = Profit

• 12 x 13000 - (1000 + 12 x 10000) = X
• 156000 - 121000 = X
• 35000 = X

• 14 x 12700 - (1000 + 14 x 10000) = X
• 177800 - 141000 = X
• 36800 = X

• 16 x 12400 - 161000 = X
• 198400 - 161000 = X
• 37400 = X

• 18 x 12100 - 181000 = X
• 217800 - 181000 = X
• 36800 = X

Therefore, he should charge $12,400 per car to maximize his profit.

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