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A tunnel on a bike trail is being designed similar to the picture below. 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. 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.

Answer :

we have that

[tex]y=-x^2+36[/tex]

we know that

Is the equation of a vertical parabola open down

so

the vertex is a maximum

step 1

convert the equation of the parabola in the vertex form

[tex]y=-x^2+36[/tex]

[tex]y-36=-x^2[/tex]

the vertex (h,k) is the point (0,36)

Part a) The point on the graph where the height of the tunnel is a maximum is (0,36)

Part b) The points on the graph where the height of the tunnel is zero feet is when y=0

so

for y=0

[tex]-x^2+36=0[/tex]

[tex]x^2=36[/tex]

[tex]x=(+/-)6[/tex]

the points are (-6,0)  and (6,0)

see the attached figure

View image АНОНИМ

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