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=−w^2+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

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

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

(0,0)
and (60,0)

Answer 1

To identify the x-intercepts of the function \( A = -w^2 + 60w \), we need to set the area \( A \) equal to 0 and solve for the width \( w \).

So, we have:

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

Factoring out \( w \):

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

This gives us two solutions:

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

The x-intercepts, where the area \( A = 0 \), are therefore at:

- \( (0, 0) \) when \( w = 0 \)
- \( (60, 0) \) when \( w = 60 \)

Thus, the correct response is:

(0,0) and (60,0)

So the final answer is **(0,0) and (60,0)**.