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)

Question

Responses

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

(60,0)
left parenthesis 60 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 quadratic function given by the formula \( A = -w^2 + 60w \), we need to set the area \( A \) equal to zero and solve for \( w \):

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

This equation can be factored:

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

Setting each factor equal to zero gives us the possible values for \( w \):

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

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

The correct response is:

(0,0) and (60,0)