A manufacturer has been selling 1000 television sets a week at $450 each. A mark

Question

A manufacturer has been selling 1000 television sets a week at $450 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of sets sold will increase by 100 per week. Round your answers to the nearest dollar.

(a) Find the demand function (price as a function of units sold).
p(x) =

(b) How large a rebate should the company offer the buyer in order to maximize its revenue?
$  

(c) If the company experiences a cost of C(x) = 74,000 + 130x, how should the manufacturer set the size of the rebate in order to maximize its profit?
$

Explanation / Answer

a)

(1000, 450) and (1100, 440)

Slope = (440 -450)/(1100 -1000) = -1/10

p - 450 = (-1/10)(x-1000)

p -450 =(-1/10)x +100

p(x) =(-1/10)x +550

b)

Find revenue as

R(x) =x*p(x)

=(-1/10)x² +550x

Now to maximize, find derivative and set to 0

(-1/10)(2x)+550 =0

-x/5 +550=0

x/5 =550

x=2750

hence

p(2750)=(-1/10)*2750 +550 =275

rebate =$450 -275 =$175

c)

Find profit as

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

=((-1/10)x² +550x)-(74000 + 130x)

=(-1/10)x² + 420 x - 74000

Now to maximize, find derivative and set to 0

P'(x) =0

(-1/10)(2x) + 420 =0

-x/5 +420=0

x/5 =420

x =5*420

x=2100

p(2100)=(-1/10)*2100 +550 =340

rebate =$450 -340 =$110

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.