Nora is creating a dog run in her backyard for her dog, Max. She has enough material to create a run with perimeter of 120 feet. This is represented by the function
A = -w^2 + 60w, where A is the area of the run and w is the width. Which of the following BEST analyzes the meaning of the x-intercept?
A. The dog run must have a width between 0 and 900 feet.
B. The dog run must have a width between 0 and 30 feet.
C. The dog run must have a width between 0 and 120 feet.
D. The dog run must have a width between 0 and 60 feet.
The x-intercept of a function is the value of x when the graph of the function intersects the x-axis. In this case, the x-intercept represents the width of the dog run when the area is 0.
To find the x-intercept, we set the area A equal to 0 and solve for w:
0 = -w^2 + 60w
Simplifying the equation, we get:
w^2 - 60w = 0
Factoring out w:
w(w - 60) = 0
Setting each factor equal to 0:
w = 0 or w - 60 = 0
w = 0 is not a valid width for the dog run, so we disregard it. The valid width for the dog run is w = 60.
Therefore, the x-intercept represents the valid width for the dog run, which is 60 feet.
So, the answer is D. The dog run must have a width between 0 and 60 feet.