How might the integral be used to in business?

Question

How might the integral be used to in business?

Answer 1

Lets say you want to sell a number of products.  A certain number of products will get you a maximum profit.  If you know your maximum profit, you can use that information to create a derivative equation that equals zero.  Then you would find the anti-derivative of that to find your profit for each amount of products sold.

 

 

For example:

 

Maximum profit is $500 at 20 chocolate bars are sold.

 

Create a derivative equation in terms of x that equal zero.  When you solve for x, you should get 20.

 

-200(x - 20)(x + 10) = 0

 

 

Expand the left side of the equation.

 

-200(x² - 10x - 200) = 0

 

-200x² + 2000x + 40000 = 0

 

 

Next, find the anti-derivative of the left side.

 

P(x) = -66.67x³ + 1000x² + 40000x + C

 

where C is any constant.

 

 

The value of P(x) is the profit earned for each number of chocolate bars sold.

 

Finally, we solve for C by plugging in x=20 and P(x)=500

 

500 = -66.67(20)³ + 1000(20)² + 40000(20) + C

 

500 = 666640 + C

 

- 666140 = C

 

 

P(x) = -66.67x³ + 1000x² + 40000x - 666140