Question

the height of the new bike tunnel, y, in feet, is given by the following equation, where x is the horizontal distance, in feet, from the center line of the opening of the tunnel.

$y = -x^2 + 36$

note that negative x - values indicate the distance to the left of the center line, and positive x - values indicate the distance to the right of the center line.

plot the points on the graph where the height of the tunnel is a maximum and where the height of the tunnel is zero feet.

Explanation:

Step1: Find maximum height

The equation $y = -x^2 + 36$ is a downward-opening parabola. Its vertex (maximum point) occurs at $x=0$. Substitute $x=0$: $y = -(0)^2 + 36 = 36$. So maximum at (0,36).

Step2: Find zero height

Set $y=0$: $0 = -x^2 + 36$. Solve for $x$: $x^2 = 36$ → $x = \pm 6$. So zero height at (6,0) and (-6,0).

Answer:

Maximum height at (0, 36); Zero height at (6, 0) and (-6, 0)