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

Answer 1

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

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

Rearranging this gives:

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

Factoring out a \( w \):

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

Setting each factor to zero gives us the solutions:

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

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

The correct response identifying the x-intercepts is:

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