Chapter 9.1, Problem 94E

To determine

To find the width of lattice arch at the ground level

Answer to Problem 94E

The width of lattice arch at the ground level $\approx 5.657$

Explanation of Solution

Given information :

Height of parabolic lattice arch at vertex = 8 feet

Width of lattice arch at height of 4 feet = 4 feet

  

Concept used:

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

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

From the given figure, vertex is located at the coordinates $\left(0,8\right)$ and the endpoint is at coordinates $\left(2,4\right)$

From above clearly seen

  $\left(h,k\right)\to (0,8)\text{ and }\left(x,y\right)\to \left(2,4\right)$

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

  $\begin{array}{l}{\left(2-0\right)}^2=4p\left(4-8\right)\\ 4=4p\times \left(-4\right)\\ 4=-16p\\ p=\frac{-4}{16}\\ p=\frac{-1}{4}\end{array}$

Getting equation of parabola by putting values of $\left(h,k\right)$ and $p$ in standard equation of parabola

  $\begin{array}{l}{\left(x-0\right)}^2=4\left(\frac{-1}{4}\right)\left(y-8\right)\\ x^2=-\left(y-8\right)\end{array}$

The equation that describes the parabola is

  $x^2=-\left(y-8\right)$

At ground level $y=0$

Therefore,

  $\begin{array}{l}{x}^2=-\left(0-8\right)\\ x^2=8\\ \text{taking square root both sides}\\ \text{x=}\sqrt{8}=2\sqrt{2}\end{array}$

This value of x is half the distance of the width of the lattice arch.

Hence,

The width of lattice arch at the ground level $=2x=2\times 2\sqrt{2}=4\sqrt{2}\approx 5.657feet$