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

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

(0,0)
left parenthesis 0 comma 0 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 \( A = 0 \) and solve for \( w \):

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

Factoring out \( w \):

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

Setting each factor to zero gives us:

1. \( w = 0 \)
2. \( -w + 60 = 0 \) which simplifies to \( w = 60 \)

Thus, the x-intercepts occur at the points where \( A = 0 \):

• When \( w = 0 \), \( A = 0 \): This represents the point \( (0, 0) \).
• When \( w = 60 \), \( A = 0 \): This represents the point \( (60, 0) \).

Therefore, the correct answer is:

(0,0) and (60,0)

So the final response would be:

\( \text{(0,0) and (60,0)} \)