Chapter 9.1, Problem 95E

To determine

To find the equation of the parabola

Answer to Problem 95E

The equation that describes the parabola is

  $y^2=640x$

Explanation of Solution

Given information : 

Road engineers design a parabolic entrance from a straight street to an interstate highway

  

Concept used:

The standard equation of a parabola that has horizontal axis of symmetry is:

  ${\left(y-k\right)}^2=4p\left(x-h\right)\left(where\left(h,k\right)\text{ is vertex of parabola}\right)$

Here $\left(h,k\right)\to \left(0,0\right)$

Therefore,

Standard equation reduces to

  $y^2=4px$

Explanation: 

From the given figure, vertex is located at the coordinates $\left(0,0\right)$ and the endpoint is at coordinates $\left(1000,800\right)$

From above clearly seen

  $\left(h,k\right)\to (0,0)\text{ and }\left(x,y\right)\to \left(1000,800\right)$

Substituting values of endpoints in standard equation of parabola to find value of p

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

Getting equation of parabola by putting value of $p$ in standard equation of parabola

  $\begin{array}{l}{y}^2=4\times 160\times x\\ y^2=640x\end{array}$

Therefore, 

The equation that describes the parabola is

  $y^2=640x$