# 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...

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?

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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

=> 194.096

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.**