Question

Q.6(b) A company estimates that the demand for its product fluctuates with the

price it charges. The demand function is (05 Marks)

q= 280,000 – 400p

Where q equal the number of units demanded and p equals the price in dollars.

The cost of producing q units of the product is estimated by the function

C = 350,000 + 300q + 0.0015q2

i. Determine the number of units that should be produce in order to

maximize the annual profit.

ii. What price should be charged?

iii. What is the annual profit expected to equal?

Answer 1

$$Profit=Revenue-Cost$$ $$q = 100000 - 200p=>p(q)=500-0.005q$$ $$R(q)=p(q)\cdot q=500q-0.005q^2$$ $$P(q)=R(q)-C(q)=$$ $$=500q-0.005q^2-(150000 + 100q + 0.003q^2)=$$ $$=400q-0.008q^2-150000$$

a.

$P'(q)=400-0.016q$

Find the critical number(s)

$P'(q)=0=>400-0.016q=0=>q=25000$

$P''(q)=-0.016<0$

The function $P(q)$ has a local maximum at $q=25000.$

Since the function $P(q)$ has the only extremum, then the function $P(q)$ has the absolute maximum at $q=25000.$

A company has to produce $q=25000$ units in order to maximize annual profit.

b.

$$p(25000)=500-0.005(25000)=375$$

The price is Rs375.

c.

$$P(25000)=400(25000)-0.008(25000)^2-150000=$$ $$=4850000$$

The expected annual profit is Rs4,850,000.