6 pts) A manufacture has been selling 1900 television sets a week at $540 each.

Question

6 pts) A manufacture has been selling 1900 television sets a week at $540 each. A market survey indicates that for each $11 rebate offered to a buyer, the number of sets sold will increase by 110 per week. a) Find the function representing the demand p(x), where x is the number of the television sets sold per week and p(x) is the corresponding price. p(x)= b) How large rebate should the company offer to a buyer, in order to maximize its revenue? dollars c) If the weekly cost function is 171000+180x, how should it set the size of the rebate to maximize its profit?

Explanation / Answer

Let’s suppose that I don't know what the demand function is, but if x = the number of additional units per week, then the revenue will be;

R = (1900 + x)(540 - (x/10))
R = 1026000 -190x + 540x - x^2/10
R = 1026000 + 350x - x^2/10

Solve this using a calculator to do the quadratic equation. If you are doing calculus, take the derivative and set to zero. The answer will be x = 175 which means the rebate should be $175 per TV.

Profit = Revenue - Costs. If costs = 171000 + 180 x the number of units made, then;

P = (1900 + x)(540 - x/10) - (171000 - 180(1900 + x))
P = 1026000 -190x + 540x - x^2/10 -171000 +342000 + 180x
P=1197000 + 530x - x^2/10

Take the derivative and set it to zero
0 = 530 - x/5
x = 2650

Since the rebate is x/10, the rebate should be 265 in order to maximize profit. If you ever wondered how the manufacturers could afford large rebates and make money, often it is because their costs are reduced through improved efficiency and reduced material cost through volume and it can pay off.

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