A company manufactures and sells x cellphones per week. The weekly price-demand

Question

A company manufactures and sells x cellphones per week. The weekly price-demand and cost equations are given below.

p=600-0.1x and C(x)=15,000+135x

(A) What price should the company charge for the phones, and how many phones should be produced to maximize the weekly revenue? What is the maximum weekly revenue?

The company should produce_____phones each week at a price of $_____.(Round to the nearest cent as needed.)

(B) What price should the company charge for the phones, and how many phones should be produced to maximize the weekly profit? What is the maximum weekly profit?

The company should produce_____phones each week at a price of $_____.(Round to the nearest cent as needed.)

The maximum weekly profit is $____.(Round to the nearest cent as needed.)

Explanation / Answer

A.
p = 500 - 0.1x
p is the price per unit
revenue = quantity * price/unit
R(x) = revenue = p(x)*x = 500x - 0.1x ²

p(x) maximum when first derivative is set to 0
500 - 0.2x = 0 ==> x = 500/0.2 = 2500 quantities
price/unit : p = 500 - 0.1*2500 = 500 - 250 = 250
revenue :
r(2500) = 500*2500 - 0.1*2500 ²
r(2500) = 2500(500 - 250) = 625000
The company should produce 2500 phones each week at a price of $250
The maximum weekly revenue is $625000
------------------
B.
Profit = revenue - cost
Profit = 500x - 0.1x ² - 15000 - 140x
Profit = 360x - 0.1x ² - 15000

profit max ==> 360 - 0.2x = 0 ==> x= 1800(phones to produce)
price/unit : 500-180 = $320
maximum profit = 360*1800 - 0.1*1800 ² - 15000 = 309000

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