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REY [17]
3 years ago
11

A manufacturer has been selling 1000 television sets a week at $450 each. A market survey indicates that for each $10 rebate off

ered 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 + 150x, how should the manufacturer set the size of the rebate in order to maximize its profit?
Mathematics
1 answer:
Slav-nsk [51]3 years ago
4 0

Answer:

a) P(x) = 550 - x/10

b) $175

c) $100

Step-by-step explanation:

Number of television sold per week = 1000

The selling price for each = $450

a) If $10 rebate offered to the buyer increases the sale by 100 per week, the price will decrease by

1/100*10 = 1/10 per unit

Let X be the number of television sold per week

The increase in number sold per week = x - 1000

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

P(x) = 450 - x/10 + 100

= 550 - x/10

The demand function is

P(x) = 550 - x/10

b) Let R(x) be the revenue

R(x) = xP(x)

= x[550 - x/10]

= 550x - x^2/10

Differentiate R(x) with respect to x

R'(x) = 550 - 2x/10

= 550 - x/5

R'(x) is the marginal revenue

The revenue is maximized when R'(x) = 0

550 - x/5 = 0

550 = x/5

x = 550*5

x = 2750

Recall that P(x) = 550 - x/10

Put x= 2750

P(2750) = 550 - 2750/10

= 550 - 275

= 275

The rebate to maximize the revenue will be 450 - 275 = $175

c) C(x) = 74000 + 150x

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

P(x) = 550x - x^2/10 - (74000 + 150x)

= 550x - x^2/10 - 74000 - 150x

Collect like terms

P(x) = 550x - 150x - x^2/10 - 74000

= 400x - x^2/10 - 74000

Differentiate P(x) with respect to x

P'(x) = 400 - 2x/10

= 400 - x/5

P'(x) is the marginal profit.

The profit is maximum when P'(x) = 0

400 - x/5 = 0

400 = x/5

x = 400*5

x = 2000

Recall that P(x) = 550 - x/10

P(2000) = 550 - 2000/10

= 550 - 200

= 350

The rebate to maximize the revenue = 450 - 350

= $100

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