cost function equation

Question

A small firm observed that as the number of units of its product increases from 5000 to 7500,the total cost of production increases from $20,000 to $25,000. It was further observed that revenue function of the product was R=5x-1000.
a)Determine the cost function in order to show the relationship between the cost (y) and the number of units (x) made assuming linear relationship.
b)Determine the profit function for the product and its profit at x=6000.

Answer 1

To find the cost function we can use the high-low method. The slope of the cost line is

 

b = ($25,000 - $20,000) / 7,500 - 5,000)

b = $5,000 / 2,500 units

b = $2 per unit

 

Now we can use the high or low numbers to find the intercept a

 

Total cost Y = a + bX

Y = a + $2*7,500

$25,000 = a + $15,000

a = $10,000

 

Revenue = $5X - $1,000

Revenue = Fixed Cost + Variable cost * units + Profit

$5*6,000 - $1,000 = $10,000 + $2*6,000 + Profit

$29,000 = $10,000 + $12,000 + Profit

$29,000 - $22,000 = $7,000 profit at a volume of 6,000 units.

 

 

Answer 2

a)

When the number of units(x) is 5000, the cost(y) is 20000

The equation of a straight line is:

y = mx + b

20000 = 5000m + b   -------> b = 20000-5000m

 

When the number of units(x) is 7500, the cost(y) is 25000

25000 = 7500m + b

25000 = 7500m + 20000 - 5000m

5000 = 2500m

m = 2

b = 10000

The cost function is y = 2x + 10000

 

b) Profit = Revenue - Cost

Profit = 5x-1000-2x-10000

Profit = 3x-11000

Profit = 3*6000-11000 = $7000