Question

Luca is building a garden in his backyard. He needs to erect a fence to keep his dog Luna from digging in it. He has enough materials for a fence with 120 feet around the perimeter. This function is represented by the formula A=−w2+60w
, where A
is the area and w is the width. Which of the following correctly identifies the x-intercept(s)?(1 point)
Responses

(60,0)
and (30,900)
left parenthesis 60 comma 0 right parenthesis and left parenthesis 30 comma 900 right parenthesis

(0,0)
left parenthesis 0 comma 0 right parenthesis

(0,0)
and (60,0)
left parenthesis 0 comma 0 right parenthesis and left parenthesis 60 comma 0 right parenthesis

(60,0)

Answer 1

To find the x-intercept(s) of the function \( A = -w^2 + 60w \), we need to set the area \( A \) equal to zero and solve for \( w \). The x-intercepts occur where the graph (in this case, the area as a function of width) intersects the x-axis, which is where \( A = 0 \):

\[
-w^2 + 60w = 0
\]

Factoring out \( w \):

\[
w(-w + 60) = 0
\]

Setting each factor equal to zero gives us:

1. \( w = 0 \)
2. \( -w + 60 = 0 \) → \( w = 60 \)

So, the x-intercepts are at \( (0, 0) \) and \( (60, 0) \).

Therefore, the correct response is:

**(0,0) and (60,0)**.