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
(0,0)
left parenthesis 0 comma 0 right parenthesis
(60,0) and (30,900)
left parenthesis 60 comma 0 right parenthesis and left parenthesis 30 comma 900 right parenthesis
(60,0)
left parenthesis 60 comma 0 right parenthesis
(0,0) and (60,0)
To find the x-intercepts of the function A = -w^2 + 60w, we need to determine the values of w for which A = 0. The x-intercepts represent the points on the graph where the curve touches the x-axis.
Let's set A to 0 and solve for w:
0 = -w^2 + 60w
To solve this quadratic equation, we can factor out w:
w(-w + 60) = 0
This gives us two solutions:
w = 0 or -w + 60 = 0
For the second solution, we solve for w:
w = 60
These solutions give us the two x-intercepts for the function, which correspond to the width (w) of the garden when the area (A) is zero:
(0,0) and (60,0)
The correct answer to identify the x-intercepts is:
(0,0) and (60,0)