Chapter 1.6, Problem 50E

To determine

To find: The height of the flagpole to the nearest inch.

Answer to Problem 50E

The height of the flagpole is 32 ft 4 inch.

Explanation of Solution

Given:

The length of each wire is 5 ft longer than the height of pole.

The distance between the wires fixed on the ground is equal to the length of one wire.

Calculation:

Let x be the height of flagpole.

Tabulate the given information algebraically.

In words	In Algebra
Height of flagpole	x
Length of wire	$x+5$
Distance between pole and a guy wire	$\frac{x+5}{2}$

Table (1)

Model the equation for the above information as follows.

From the given data, there is an isosceles triangle appear here.

Use Pythagoras theorem to find the length of the flagpole as follows.

$\begin{array}{l}{\left(\text{Distance}\text{between}\text{pole}\text{and}\text{wire}\right)}^2+{\left(\text{Length}\text{of}\text{pole}\right)}^2={\left(\text{Length}\text{of}\text{wire}\right)}^2\\ {\left(\frac{x+5}{2}\right)}^2+x^2={\left(x+5\right)}^2\\ {\left(x+5\right)}^2+4x^2=4{\left(x+5\right)}^2\\ 4x^2=3{\left(x+5\right)}^2\end{array}$

On further simplification,

$\begin{array}{l}4x^2=3\left(x^2+25+10x\right)\\ 4x^2=3x^2+75+30x\\ x^2-30x-75=0\end{array}$

Use quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ to find the value of x.

$\begin{array}{c}x=\frac{-\left(-30\right)\pm \sqrt{{\left(-30\right)}^2-4\times 1\times \left(-75\right)}}{2\times 1}\\ =\frac{30\pm \sqrt{900+300}}{2}\\ =\frac{30\pm \sqrt{1200}}{2}\\ =\frac{30\pm 34.64}{2}\end{array}$

On further simplification,

$\begin{array}{l}x=\frac{30+34.64}{2},x=\frac{30-34.64}{2}\\ x=32.32,x=-2.32\end{array}$

The height of the pole cannot be negative.

So the height of the pole, $x=32.32$.

Thus, the height of the flagpole is 32 ft 4 inch.