Correct question is;
A manufacturer has been selling 1400 television sets a week at $450 each. A market survey indicates that for each $21 rebate offered to a buyer, the number of sets sold will increase by 210 per week
A) Find the demand function (price p as a function of units sold x).
p(x)=______
(B) How large a rebate should the company offer the buyer in order to maximize its revenue?
$=_____
Answer:
A) p(x) = (-1/10)x + 590
B) Rebate = $170
Explanation:
A) We are told that p(x) is the demand function and x is the number of TV sets sold per week.
Now, since for each $21 rebate offered, the number of sets sold increases by 210 per week, it means that the slope here of this demand function is; m = -21/210 = -1/10
Now, he has been selling 1400 TV sets a week at $450 each. This means; p(1400) = $450
Thus, the demand function will be;
p(x) - 450 = (-1/10)(x - 1400)
Expanding the RHS;
p(x) - 450 = (-1/10)x + 140
Add 450 to both sides to get;
p(x) = (-1/10)x + 140 + 450
p(x) = (-1/10)x + 590
B) Formula for revenue is;
R = price × quantity sold
Our demand function is p = (-1/10)x + 590
Making x the subject, we have;
x = 5900 - 10p
x is quantity sold.
Thus,
R = p(5900 - 10p)
R = 5900p - 10p²
Maximum price will occur at dR/dP = 0
Thus;
dR/dP = 5900 - 20p
At dR/dP = 0,we have;
20p = 5900
p = 5900/20
p = $280
Thus, rebate = 450 - 280 = $170