Chapter 9.2, Problem 34PPS

To determine

To write: The equation of the parabola to model the arch.

Answer to Problem 34PPS

The equation of the parabola path of the bal is $y=-\frac{1}{6}{x}^2$ .

Explanation of Solution

Given:

Assume that the origin is ay the point where the pole and umbrella meet at the vertex of the arch.

Calculation:

The vertex is at $\left(0,0\right)=\left(h,k\right)$ and focus is $\left(0,-1.5\right)$

Use the equation of directrix $y=k+\frac{1}{4a}$ to find the value of $a$ .

  $\begin{array}{c}-1.5=0+\frac{1}{4a}\\ \frac{1}{a}=-6\\ a=-\frac{1}{6}\end{array}$

Substitute $-\frac{1}{6}$ for $a$ , 0 for $h$ and 0 for $k$ in the equation $y=a{\left(x-h\right)}^2+k$ to find the equation of parabola.

  $\begin{array}{c}y=-\frac{1}{6}{\left(x-0\right)}^2+0\\ =-\frac{1}{6}{x}^2\end{array}$

Therefore, the equation of the parabola to model the arch is $y=-\frac{1}{6}{x}^2$ .