Question

Suppose a company has fixed costs of $47,600 and variable cost per unit of 4/9x + 333 dollars,

where x is the total number of units produced. Suppose further that the selling price of its product is

1767 −5/9x dollars per unit.

(a) Find the break-even points. (Enter your answers as a comma-separated list.)
x =  

  

(b) Find the maximum revenue. (Round your answer to the nearest cent.)
$  

(c) Form the profit function P(x) from the cost and revenue functions.
P(x) =  

  

Find maximum profit.
$  

(d) What price will maximize the profit? (Round your answer to the nearest cent.)
$

Answer 1

Fixed cost is 47,600;
Variable cost is 4x/9 + 333
Selling price per units = 1767-5x/9 per unit
For break even: Fixed cost + variable cost /unit * number of units = selling price* number of units
Fixed cost = (selling price - variable cost) * number of units
47600 = (1767 - 5x/9 - 4x/9 -333) * x
x ( 1434 - x) = 47600
x² - 1434 x + 47600 =0;
x= 1400; x=34;
Break even points = 34 units and 1400 units;

b) Maximum revenue = (1767-5x/9) (x) = 1767x - 5x²/9;
For maximum revenue, dR/dx =0 so
1767 - 10x/9 =0;
x= 1767*9/10 = 1590.3~= 1590; Since number of unit sold cant be a decimal;
If x=1590, then total revenue = 1590 ( 1767-5(1590)/9) = 1590 ( 883.66) =1,405,030$

c) Profit function = (SP - CP) *number of units
= (1767-5x/9)x - (47600 + x(4x/9 + 333)) = 1767x - 5x²/9 - 47600 -4x²/9 -333x
= -x² + 1434x -47600  

For maximum profit, dP/dx =0;
-2x + 1434=0
x= 1434/2= 717;
At x=717, P(x) = - 717² + 1434(717) - 47600 = 4,66,489$
Thus, maximum profit occurs at 717 units which is = 466,489$

d) Price that maximises the profit is :
1767 - 5(717)/9 = 1767 - 398.33 = 1368.67$