Question

A parabolic arch has a height of 25 feet and a span of 40 feet. How high is the arch 8 feet from each side of the center?

10 ft.

16 ft.

21 ft.

Answer 1

The correct option is;

21 ft

Explanation:

The equation of the parabolic arc is as follows;

y = a(x - h)² + k

Where the height is 25 ft and the span is 40 ft, the coordinates of the vertex (h, k) is then (20, 25)

We therefore have;

y = a(x - 20)² + 25

Whereby the parabola starts from the origin (0, 0), we have;

0 = a(0 - 20)² + 25

0 = 20²a + 25 → 0 = 400·a + 25

∴a = -25/400 = -1/16

The equation of the parabola is therefore;

`y = (-(1)/(16))(x-20)^2 + 25`

To find the height 8 ft from the center, where the center is at x = 20 we have 8 ft from center = x = 20 - 8 = 12 or x = 20 + 8 = 28

Therefore, plugging the value of x = 12 or 28 in the equation for the parabola gives;

`y = (-(1)/(16))(12-20)^2 + 25 = (-(1)/(16))(-8)^2 + 25 = 21 \ ft`.