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

Answer 1

To find the x-intercepts of the function \( A = -w^2 + 60w \), we need to determine where the area \( A \) is equal to zero. This means we solve the equation:

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

Factoring out \( w \):

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

Setting each factor to zero gives us:

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

Thus, the x-intercepts are the points where \( A = 0 \), which corresponds to \( (0, 0) \) and \( (60, 0) \).

So the correct response is:

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