Question

The price P per unit at which a company can sell all that it produces is given by the function P(x) = 300 — 4x. The cost function is c(x) = 500 + 28x where x is the number of units produced. Find x so that the profit is maximum.

Answer 1

total selling price = P.x
= (300 - 4x)x
profit = P*x - C
= (300 - 4x)x - (500 +28x)

for this to be maximum, (300 - 4x)x should be maximum

maximum value of a quadratic equation = -b/2a
= -300/(-2*4)
= 37.5

this can yield two values 37 and 38. you will find that P.x is same for both value

so now C is less for 37
so the answer is 37

Answer 2

Concept:

Profit means the difference between the selling price and the cost price of a firm.

Given:

We are given that:

The selling price function is:

P(x) = x( 300 - 4 x).

The cost function is :

c(x) = 500 + 28 x

Find:

We need to find x so that the profit is maximum.

Solution:

The selling price function is:

P(x) = x( 300 - 4 x).

The cost function is :

c(x) = 500 + 28 x

Profit function will be:

selling price function - cost function

=300x - 4 x² - (500 + 28 x)

=300 x - 4 x² - 500 - 28 x

P (x) =-4x² +272 x - 500

We will differentiate it with respect to x:

P' (x)=-8x+272.

Put it equal to 0:

8x=272

x=34.

Therefore, x=34, if the profit has to be maximum.

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