Question

5. Suppose the cost of publishing a certain book is Birr 10,000 to set up the

annual press run, plus Birr 8 for each book actually printed. The publisher sold
7,000 copies last year at Birr 13 each, but this year sales dropped to 5,000
copies when the price was raised to Birr 15 per copy. Assuming that as many as
10,000 copies can be printed in a single press run find the number of copies that
should be printed in order to maximize the year's profit. What is maximum profit
and what should be the selling price of each copy?

Answer 1

Explanation:

Let's call the number of books printed this year "x".

The profit for this year can be represented as:

Profit = (Revenue from book sales) - (Cost of printing)

Revenue from book sales = x * selling price

Cost of printing = 10,000 + 8 * x

So, Profit = x * selling price - (10,000 + 8 * x) = (selling price - 8) * x - 10,000

To maximize the profit, we need to find the value of x and the selling price that maximizes the expression (selling price - 8) * x - 10,000.

Taking the derivative of the expression with respect to x, we get:

d(Profit)/dx = (selling price - 8)

Setting this equal to 0 and solving for the selling price, we get:

selling price = 8

So, the maximum profit is obtained when the selling price is 8 Birr and x is such that the total cost of printing is equal to the total revenue from book sales.

Substituting x = 10,000 and the selling price = 8 Birr into the equation for profit, we get:

Profit = (8 * 10,000) - (10,000 + 8 * 10,000) = 80,000 - 18,000 = 62,000 Birr

Therefore, the maximum profit is Birr 62,000 and the optimal selling price is Birr 8.