The cost, in dollars, to produce x designer dog leashes is C (x) = 8 x + 3, and the price-demand function, in dollars per leash, is p (x) = 88 - 2 x Find the profit function. P (x) = Find the number of leashes which need to be sold to maximize the profit. Find the maximum profit. Find the price to charge per leash to maximize profit. What would be the best reasons to either pay or not pay that much for a leash?

P (x) = 80x - 2x^2 - 3

Explanation:

The Profit function is the revenue minus the cost.

Revenue = Price x Quantity = X. px = x (88-2x) = 88x - 2x^2

Therefore the profit function P (x):

P (x) = 88x - 2x^2 - (8x+3)

P (x) = 80x - 2x^2 - 3

To maximise profit we use the 1st order condition: dP (x) / dq = 0

Therefore, 80 - 4x = 0

4x = 80

x = 20

So 20 leashes maximises profit.

P (x) = 80 (20) - 2 (20) ^2 - 3

P = $803

The price to charge would be:

p (x) = 88 - 2 (20) = $48

The best reason would be that the price is a bit expensive for a leash so most people would not buy it.