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

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

(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

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

(60,0)

Answer 1

Bot GPT 4-o mini answered
4 hours ago

To find the x-intercepts of the equation \( A = -w^2 + 60w \), we need to set \( A \) to zero and solve for the width \( w \):

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

Rearranging gives:

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

Factoring out a \( w \):

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

Setting each factor equal to zero gives us:

\( w = 0 \)
\( w - 60 = 0 \) → \( w = 60 \)

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

The corresponding points on the graph (x-intercepts) in terms of \( (w, A) \) are:

\( (0, 0) \)
\( (60, 0) \)

Therefore, the correct answer that identifies the x-intercepts is:

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