Question

An arch is in the shape of a parabola with its vertex at the top. It has a span of 100 feet and a maximum height of 30 feet. Find the equation of the parabola, and determine the height of the arch 35 feet from the center of the base of the arch.

Answer 1

Answer:

The equation of parabola is $y=-\frac{3}{250}x^2+30$, where origin is the center of base. The height of the arch 35 feet from the center of the base of the arch is 15.3 feet.

Step-by-step explanation:

The vertex form of a parabola is

$y=a(x-h)^2+k$          ... (1)

where, (h,k) is vertex and a is a constant.

Let origin be the center of base.

It is given that the arch is a parabola, it has a span of 100 feet and a maximum height of 30 feet. It means the vertex of the parabola is (0,30) and the parabola passes through the points (-50,0) and (50,0).

Substitute h=0 and k=30 in equation (1).

$y=a(x-0)^2+30$            ....(2)

$y=ax^2+30$

The parabola passes through the point (0,50).

$0=a(50)^2+30$

$-30=2500a$

$-\frac{30}{2500}=a$

$-\frac{3}{250}=a$

Substitute $a=-\frac{3}{250}$ in equation (2).

$y=-\frac{3}{250}x^2+30$

Substitute x=35 to find the height of the arch 35 feet from the center of the base of the arch.

$y=-\frac{3}{250}(35)^2+30$

$y=-14.7+30$

$y=15.3$

Therefore, the equation of parabola is $y=-\frac{3}{250}x^2+30$, where origin is the center of base. The height of the arch 35 feet from the center of the base of the arch is 15.3 feet.