Question

Acme Tools manufactures and sells cordless in a market where price (p) and demand (D) are related as follows:

p = $40 + (3,000)/D – (4,800)/D²

 

The fixed cost (C_F) is $1000 per month and the variable cost per drill (c_v) is $46.

 

a)     How many drills should be produced each month to maximize profits?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b)     How much is the maximum profit each month?

 

 

Answer 1

Given;

$P = \$40 + \frac{3,000}{D} – \frac{4,800}{D^2}.................................(i)$

$FC=\$1000$ and $VC=\$46$ per drill, $TC=\$1000+46D$

(a) Number of drills to be produced each month to maximize profits

Profit is maximized at the point where MR=MC

$TR=P\times D$

$=[40+\frac{3000}{D}-\frac{4800}{D^2}]\times D$

$=40D+3000-\frac{4800}{D}$

$MR=40-(-\frac{4800}{D^2})$

$=40+\frac{4800}{D^2}$

TC=$1000+46D

$MC=46$

Taking MR=MC,

$40+\frac{4800}{D^2}=46$

$\frac{4800}{D^2}=46-40$

$\frac{4800}{D^2}=6$

$4800=6D^2$

$800=D^2$

$D=28.2843\approx28$

(b)

$Profit=TR-TC$

Substituting D in equation (i) we calculate the price as follows:

$P = \$40 + \frac{3,000}{28} – \frac{4,800}{28^2}$

$P=141.02$

$Profit=(P\times D)-(TC)$

$=(141.02\times 28)-[1000+46(28)]\\3,948.56-2,288=1,660.56$

Therefore the maximum profit per month is $1,660.56