Question

Research shows that the demand function for a new product is d(x)=-5 x+18, where x represents the number of items in thousands and d represents the item price in $.The cost function is C(x)=2 x+9.

a. State the Revenue function R(x) [1 mark]

b. Find the corresponding Profit function P(x). [1 mark]

c. How many items must be sold to maximize profit? [2 marks]

c. How many items must be sold for the company to break even? [2 marks] Hint: R(x)=x · p(x). The profit function is the difference P(x)=R(x)-C(x)

Answer 1

SOLUTION :

a. 

Demand function given :

d(x) = - 5x + 18 

Where d is price and x is quantity demanded in thousands.

We can say, demand function is :

p(x) = - 5x + 18 

So, 

R(x) 

= p(x) * x 

= (- 5x + 18) * x

= - 5x^2 + 18x (ANSWER).

b.

Profit function :

P(x) 

= R(x) - C(x)

= (- 5x^2 + 18x) - (2x + 9)

= - 5x^2 + 16x - 9 (ANSWER).

c.

For maximum profit, 

dP/dx = 0 

=> d/dx (- 5x^2 + 16x - 9) = 0

=> - 10x + 16 = 0

=> x = 16/10 = 1.6 thousands = 1600 units 

Items to be sold = 1600 units (ANSWER).

d.

For break even, P = 0

=> - 5x^2 + 16x - 9 = 0

=> 5x^2 - 16x + 9 = 0 

=> x = 16/10 +/- sqrt(16^2 - 4(5)(9))/10

=> x = 1.6 +/- 0.8718

=> x = 0.7282 , 2.4718   (thousands)

=> x = 728 , 2472 

So, for BEP, 728 units or 2472 units are to be sold . (ANSWER).

Answer 2

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