Answer:
The equation of parabola is [tex]y=-\frac{3}{250}x^2+30[/tex], 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
[tex]y=a(x-h)^2+k[/tex] ... (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).
[tex]y=a(x-0)^2+30[/tex] ....(2)
[tex]y=ax^2+30[/tex]
The parabola passes through the point (0,50).
[tex]0=a(50)^2+30[/tex]
[tex]-30=2500a[/tex]
[tex]-\frac{30}{2500}=a[/tex]
[tex]-\frac{3}{250}=a[/tex]
Substitute [tex]a=-\frac{3}{250}[/tex] in equation (2).
[tex]y=-\frac{3}{250}x^2+30[/tex]
Substitute x=35 to find the height of the arch 35 feet from the center of the base of the arch.
[tex]y=-\frac{3}{250}(35)^2+30[/tex]
[tex]y=-14.7+30[/tex]
[tex]y=15.3[/tex]
Therefore, the equation of parabola is [tex]y=-\frac{3}{250}x^2+30[/tex], 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.
