Question

A bookstore's monthly earnings may be modeled by the function f(x)=−3x2+12x+180
, where f(x)
represents the amount of money the bookstore earns in a month and x
represents the number of months that have passed since the bookstore opened.

Which statements are true? Select three that apply.

(3 points)
Responses

The zeros of the function represent the months in which the bookstore earned $0.
The zeros of the function represent the months in which the bookstore earned $0.

The most money the bookstore will earn during one of the months is $180.
The most money the bookstore will earn during one of the months is $180.

The most money the bookstore will earn during one of the months is $192.
The most money the bookstore will earn during one of the months is $192.

The only reasonable zero in this context is x=6
.
The only reasonable zero in this context is x is equal to 6.

The only reasonable zero in this context is x=180
.
The only reasonable zero in this context is x is equal to 180.

The only reasonable zero in this context is x=10
.

Answer 1

True statements (with brief justification):

1. The zeros of the function represent the months in which the bookstore earned $0.
- Solve -3x^2 + 12x + 180 = 0 → x^2 − 4x − 60 = 0 → x = 10 or x = −6. These x-values give f(x)=0.

2. The most money the bookstore will earn during one of the months is $192.
- Vertex at x = −b/(2a) = −12/(2·−3) = 2. f(2) = −3(4) + 12(2) + 180 = 192, the maximum.

3. The only reasonable zero in this context is x = 10.
- x = −6 is a zero but nonphysical (before opening), so x = 10 is the only realistic month with $0 earnings.