Finding quantity given revenue function

Question

The revenue from the sale of a product is R = -x^3 - 5x^2 + 250x. When 10 units are sold the revenue is$1000. Find another amount of units that can be sold to give $1000.

Answer 1

1000 = -x^3 - 5x^2 + 250x

 

0 = -x^3 - 5x^2 + 250x - 1000

 

0 = x^3 + 5x^2 - 250x + 1000

 

Since x=10 is a given solution, we can factor out (x-10)

 

Synthetic division

 

10  |        1     5    -250   1000

                     10    150   -1000

       ----------------------------------

               1   15   -100    0

 

The quadratic factor is x^2 + 15x - 100

 

   which factors as  (  x    +   20 )( x     -          5  )

 

 

x+20 = 0 ---> x = -20 and the quantity sold cannot be negative.

 

So x-5 = 0 ---> 5 units will produce $1000

 

The check is left for you, but it works.