A company has found that when x units of a product are manufactured and sold,
its revenue is given by x 2 + 100 x dollars and its costs are given by 240 x + 500 dollars. How many units must be produced and sold to make a profit of 10,000 dollars?
The profit made is equal to revenue - costs.
For x number of units the profit is given by x^2 + 100x - 240x - 500.
To make a profit of 10000, let the number of units to be produced be x.
x^2 + 100x - 240x - 500 = 10000
=> x^2 - 140x - 10500 = 0
x = [-b + sqrt (b^2 - 4ac)]/ 2a
=> [ 140 + sqrt ( 140^2 + 4*10500)]/ 2
=> [ 140 + sqrt 61600]/ 2
=> 70 + 5*sqrt 616
As the number of units cannot be fractions it has to make 195 units.
The number of units to make a profit of $10000 is 195.
Let the revenue be R, the profit be P, and the cost be C.
Then, we know that:
Profit = Revenue - cost
==> Given that R= x^2+100x
Also, given that C = 240x+500
==> P = x^2 +100x - 240x -500
==> P = x^2 -140x -500
We need to find x such that P= 10,000
==> x^2 - 140x -500 = 10,000
==> x^2 -140x -10,500 = 0
We will use the formula to find the roots.
==> x1= ( 140+248.2)/2 = 194.1 ( approx.)
Then, the number of product must be sold is 194.