Answer to Question #268310 in Calculus for toodles

Question

 The demand curve of a firm is p = 1200 - 21q and its total cost is C(q) =2q³-66q²+ 600q + 1000 where q is the output of the firm (in thousands).

(i) Derive an expression R (q ) , for the firm' s revenue function.

(ii) Derive an expression π(q) for the firm' s profit function.

(iii) Is the rate of change of profit increasing or decreasing when the output level of the firm is 10,000 units?

(iv) Determine the level of output at which profit is maximized. 

Answer 1

Solution;

Given;

$p=1200-21q$

$C(q)=2q^3-66q^2+600q+1000$

(i)

Revenue =Demand ×Output

$R(q)=(1200-21q)q$

$R(q)=1200q-21q^2$

(ii)

profit =Revenue -Total cost.

$π(q)=1200q-21q^2-2q^3+66q^2-600q-1000$

Simplies to;

$π(q)=-2q^3+45q^2+600q-1000$

(iii)

Differentiate the profit function to get the rate of change of profit.

$π'(q)=-6q^2+90q+600$

$π'(10,000)=-6(10,000)^2+90(10,000)+600=-599099400$

The rate is negative meaning that it's decreasing.

(iv)

Profit is maximised when the marginal revenue equals the marginal cost.

Marginal revenue

is found by taking the first derivative of total revenue with respect to q;

$MR=R'(q)=1200-42q$

Similarly, marginal cost is determined by taking the first derivative of the total cost function with respect to q;

$MC=C'(q)=6q^2-132q+600$

Equating MC and MR to determine the profit-maximizing quantity;

$1200-42q=6q^2-132q+600$

Resolve to;

$6q^2-90q-600=0$

Divide by 6;

$q^2-15q-100=0$

Rewrite;

$q^2-20q+5q-100=0$

$q(q-20)+5(q-20)=0$

$(q+5)(q-20)=0$

$q=-5,20$

Ignore the negative;

q=20