Algebra Two Chapter Five open-ended question: Shoe manufacturer.

Question

The profit P (in millions of dollars) for a shoe manufacturer can be modeled by P= -21x^3 + 46x is the number of shoes produced (in millions). The company now produced 1,000,000 shoes and makes a profit of $25,000,000, but would like to cut back production.

 

 

1. Write an equation that models the profit of $25,000,000.

 

 

 

2. What is the maximum amount of solutions this equation could have? Explain.

Answer 1

P = -21x^3 +46x

25 = -21x^3 +46x

21x^3 -46x + 25 = 0

x=1

21(1)^3 -46(1) +25=0

21-46 +25 = 0

x= 1 million shoes sold for a profit of $25 million

cubic equations have a maximum 3 possible real solutions. They have 3 solutions, but two may be imaginary. Either 0 or 2 imaginary solutions. imaginary solutions come in conjugate pairs.

there are two sign changes in the terms, so 2 possible positive real solutions. change x to -x and there is only 1 sign change, so only 1 possible negative real solution