Question

A manufacture has been selling 1750 television sets a week at $420 each. A market survey indicates that for each $29 rebate offered to a buyer, the number of sets sold will increase by 290 per week.

a) Find the demand function p(x), where x is the number of the television sets sold per week.
р(х) = ___________
b) How large rebate should the company offer to a buyer, in order to maximize its revenue?
c) If the weekly cost function is 122500 + 140x, how should it set the size of the rebate to maximize its profit?

Answer 1

a) The demand function is p(x) = (1750 + 290x)(420 - 29x). b) To maximize revenue, find the maximum of the demand function. c) To maximize profit, subtract the cost function from the revenue function.

Step-by-step explanation:

a) The demand function p(x) can be found by subtracting the number of rebates from the initial number of sets sold and multiplying by the price:

p(x) = (1750 + 290x)(420 - 29x)

b) To maximize revenue, we need to find the maximum value of the demand function. This can be done by finding the derivative of the demand function and setting it equal to zero:

p'(x) = 420 - 29x - 290(420 - 29x) = 0

Solving this equation will give us the value of x, which represents the number of rebates to offer.

c) To maximize profit, we need to subtract the cost function from the revenue function. The revenue function is the demand function multiplied by the price:

R(x) = (1750 + 290x)(420 - 29x)

And the profit function is:

P(x) = R(x) - C(x)

Where C(x) is the cost function. To maximize profit, we need to find the maximum value of the profit function. This can be done by finding the derivative of the profit function and setting it equal to zero:

P'(x) = R'(x) - C'(x) = 0

Solving this equation will give us the value of x, which represents the number of rebates to offer.