Chapter 9, Problem 55CLT

To determine

Find the width of the archway at ground level.

Answer to Problem 55CLT

The width of the archway at ground level is $24\sqrt{2}metres$ .

Explanation of Solution

Given:

It is given in the question that a parabolic archway is $16$ meters high at the vertex. At a height of $14$ meters, the width of the archway is $12$ meters as shown in figure given below:

  

Concept Used:

In this, use the concept of parabola and the use of standard equation ${(x-h)}^2=4p(y-k)$ .

Calculation:

Since from the figure the parabola is open downward (vertical axis),

So the standard equation is :

  ${(x-h)}^2=4p(y-k)$ ,where : $(h,k)\to (0,16)=vertex$

Solve for p using the given points $(6,14)$ :

  $\begin{array}{l}$ {(x-h)}^2=4p(y-k)$$ {(6-0)}^2=4p(14-16)$36=-8p\\ p=-4.5\end{array}$

Substituting,

  $\begin{array}{l}$ {(x-0)}^2=4(-4.5)(y-16)$x^2=-18(y-16)(Equation)\end{array}$

At $y=0$ ,

  $\begin{array}{l}{x}^2=-18(0-16)\\ x^2=288\\ x=12\sqrt{2}meters\end{array}$

This value of x is half the distance of the width of the lattice arch so multiply this by $2$ .

  $\begin{array}{l}d=2x=2(12\sqrt{2})\\ d=24\sqrt{2}metres\end{array}$

Conclusion:

The width of the archway is $24\sqrt{2}metres$ .