Question

A store has been selling 200 DVD burners sold per week at 400 each. A market survey indicates that for each $ 20 rebate offered to buyers, the number of units sold will increase by 40 a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its revenue

Answer 1

Answer:

The answer is "50".

Step-by-step explanation:

Let,

$\to p(200) = 400$

If we assume the $20 reduction would raise the number of items sold to $40 so the increment is (x - 200) if x is the number of units sold.

$\to (\frac{1}{40}) \times 20 = 0.5$

$\to D(x) = 400 - 0.5 \times ( x - 200 ) \\\\ \ and \\ \to R(x) = x \times (D(x)$

            $= x \times [ 400 - 0.5\times ( x - 200 )]\\\\ = x\times[400 - 0.5x - 100]\\\\= x\times [300 - 0.5 x]\\\\= 300x - 0.5 x^2$

$R(x)= 300x- \frac{1}{2} x^2\\\\R'(x)=300-x$

if R'(x)=0

x=300

if  0 < x  < 300  and R(x) > 0 then R(x) for maximum x is = 300

$D(300) = 400 - 0.5\times ( x - 200 )$

            $= 400 - 0.5\times ( 300- 200 )\\\\ = 400 - 150+ 100\\\\ = 500 - 150 \\\\=350$

The rebate must be  : 400 - 350  = 50