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

(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)
(
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

(60,0)

Answer 1

To find the x-intercepts of the function \( A = -w^2 + 60w \), we set the area \( A \) to zero and solve for \( w \).

So, we set the function equal to zero:

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

We can rearrange this equation:

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

Next, we factor out \( w \):

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

Setting each factor to zero gives us the solutions:

1. \( w = 0 \)
2. \( w - 60 = 0 \) which gives \( w = 60 \)

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

Based on the answer choices provided, the correct identification of the x-intercepts is:

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