Answer to Question #128014 in Calculus for ALI

Question

A company estimates that the demand for its product fluctuates with the price it charges. The demand function is given as:
q = 100,000 - 200p

Where .q. equals the number of units demanded and .p. equals the price in rupees. The total cost of producing q units of the product is estimated by the function:

C = 150,000 + 100q + 0.003q2
Required:

a.Determine how many units of q should be produced in order to maximize annual profit?
b.What price should be charged?
c.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.