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

Question

A
=
−
w
2
+
60
w
, where A
A
is the area and w is the width. Which of the following correctly identifies the x-intercept(s)?(1 point)
Responses

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

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

(0,0)
(
0
,
0
)
and (60,0)
(
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 \( A \) to 0 and solve for \( w \):

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

This can be factored:

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

Setting each factor equal to zero gives:

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

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

So the correct response that identifies the x-intercept(s) is:

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