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 = −w^2 + 60w, where 𝐴 is the area and w are the width. Which of the following correctly identifies the x-intercept(s)?

Question

A) (0,0)
B) (60,0)
C) (60,0) and (30,900)
D) (0,0) and (60,0)

Answer 1

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

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

Factoring out \( w \):

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

Setting each factor to zero gives:

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

Thus, the x-intercepts of the function are:

\[ (0, 0) \quad \text{and} \quad (60, 0) \]

Now, let's identify which answer choice correctly identifies the x-intercepts:

A) (0,0)
B) (60,0)
C) (60,0) and (30,900)
D) (0,0) and (60,0)

The correct answer is:

D) (0,0) and (60,0)