Question

The parabolic arch shown in the figure is 50 feet above the water at the center and 200 feet wide at the base.Will a boat that is 30 feet tall clear the arch 30 feet from the center?​

Answer 1

Draw a semicircle with bottom left corner at origin (0,0)

The distance across the bottom is 200 m so the right corner is (200,0)

The arch is 4 m tall so the vertex is at (100, 4)

The vertex form of the parabola is: y = a(x-h)2 + k

where vertex is (h, k)

y = a(x-100)2 + 4

Since we know the point (200, 0) is on the parabola plug in for (x, y) and solve for a:

0 = a(200 - 100)2 + 4

0 = 10000a + 4

a = -4/10000 = -1/2500

y = (-1/2500)(x-100)2 + 4 is the equation of the parabolic arch

We want to find the height of the arch where x = 20 m

y = (-1/2500)(20-100)2 + 4

y = (-1/2500)(-80)2 + 4

y = (-1/2500)(6400) + 4

y = -2.56 + 4

y = 1.44 meters height

And by the way Happy New Year 2021