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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)
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
= p(x) * x
= (- 5x + 18) * x
= - 5x^2 + 18x (ANSWER).
Profit function :
= R(x) - C(x)
= (- 5x^2 + 18x) - (2x + 9)
= - 5x^2 + 16x - 9 (ANSWER).
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).
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).
Research shows that the demand function for a new product is d(x)=-5 x+18
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