Chapter 9.1, Problem 100E

To determine

Tofind:The equation of the parabola.

Answer to Problem 100E

The equation of the parabola is $y^2=640x$ .

Explanation of Solution

Given information:

Road engineers design a parabolic entrance ramp from a straight street to an interstate highway(see figure).

  

Calculation:

The standard equation for the parabola is ${\left(y-k\right)}^2=4p\left(x-h\right)$ .

As clear from the figure that the vertex of parabola is $\left(0,0\right)$ .

Now substitute $0$ for $h$ and $0$ for $k$ in the general equation.

  $\begin{array}{c}{\left(y-0\right)}^2=4p\left(x-0\right)\\ y^2=4px\end{array}$

Now as it is clear from the figure that the parabola passes through the point $\left(1000,800\right)$ .

Now substitute $1000$ for $x$ and $800$ for $y$ in the general equation.

  $\begin{array}{c}{y}^2=4px\\ {800}^2=4p\left(1000\right)\\ 640000=4000p\\ p=\frac{640000}{4000}\\ p=160\end{array}$

Now substitute $160$ for $p$ in the equation of parabola.

  $\begin{array}{c}{y}^2=4px\\ y^2=4\left(160\right)x\\ y^2=640x\end{array}$

Therefore, the equation of the parabola is $y^2=640x$ .