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)?
• (0,0)
• (60, 0) and (30, 900)
• (60,0)
• (0, 0) and (60, 0)
The x-intercept(s) occur when the area, A, is equal to zero. Therefore, we need to solve the equation -w^2 + 60w = 0 for w.
Factor out w: w(-w + 60) = 0
Set each factor equal to zero:
w = 0 or -w + 60 = 0
For the first factor, w = 0, the x-intercept is (0,0).
For the second factor, -w + 60 = 0, rearrange to solve for w:
w = 60
So the second x-intercept is (60,0).
Therefore, the correct answer is (60,0).