Question

Need help with Trig/Pre-Calc homework Parabolas.

Answer 1

(1) Equation of the parabola:
`y=-(1)/(1368)x^2+38`

Steps:

We know the span 456 m and max height is 38 m. Since the parabola is symmetric around the vertical going through the maximum point, we know half the span, or 228 m to the left and right from the origin are the zeros of the function. We can place the parabola centered around the origin. Also, since it is an arch, the parabola is upside down, so the coefficient at x^2 is negative:

`y=-Ax^2+B`

The vertex is at (0,38), and the positive zero at (228,0). That is enough to determine A and B:

`38 = -A\cdot 0 + B\implies B=38\\0 = -A\cdot228^2 + 38 \implies A=(38)/(228^2)=(1)/(1368)\approx0.00073\\y=-(1)/(1368)x^2+38`

(2) height 204 m from the center: 7.58 m

Steps:

Height at 204 m off the center (either direction) can be calculated by plugging in x=204 into the above equation:
`y(204)=-(1)/(1368)(204)^2+38\approx 7.58`