Question

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Due to an excess of unsold units in April, the number of gaming consoles a manufacturer needs to produce in May is given by the function f(x) = x - 400, where x represents the number of new units produced in May. The manufacturer's profit per unit for May decreases due to the excess, and is given by the function g(x) = 1,200 - 2x.

Which function best describes the total profit the manufacturer can earn in May, P(x)?
A. P(x) = -x2 + 800x
B. P(x) = -x2 + 1,000x - 240,000
C. P(x) = -2x2 + 400x - 480,000
D. P(x) = -2x2 + 2,000x - 480,000

Answer 1

Option (D) is correct.

`P(x)=-2x^2+2000x-480000`

Explanation:

Given : The number of units produced in May is given by function
`f(x) = x-400`

and the manufacturer's profit per unit for May is given by the function
`g(x) = 1200-2x`

We have to determine the total profit the manufacturer can earn in May, P(x).

Consider the given data

Since, The number of units produced in May is given by function
`f(x) = x-400`

and the manufacturer's profit per unit for May is given by the function
`g(x) = 1200-2x`

Total profit that the manufacturer can earn in May is given by the product of number of units produced with the profit per unit.

that
`P(x)=F(x)\cdot G(x)`

Substitute the values, we get,

`P(x)=(x-400)\cdot(1200-2x)`

Simplify, using FOIL method
`\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd`

`=x\cdot \:1200+x\left(-2x\right)+\left(-400\right)\cdot \:1200+\left(-400\right)\left(-2x\right)`

We get,

`P(x)=-2x^2+2000x-480000`

Thus, Option (D) is correct.

Answer 2

P(x)=f(x)*g(x)
P(x)=(x-400)(1,200-2x)
P(x)=1,200x-2x^2-480,000+800x
P(x)=-2x^2+2,000x-480,000

Answer: Option D. P(x) = -2x2 + 2,000x - 480,000