Quadratic Functions from Situations Quick Check

Question

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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)
left parenthesis 60 comma 0 right parenthesis
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Answer 1

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

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

This can be factored as:

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

Setting each factor equal to zero gives:

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

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

Therefore, the correct response is:

(0,0) and (60,0) \[ \text{left parenthesis 0 comma 0 right parenthesis and left parenthesis 60 comma 0 right parenthesis} \]