A \(16.5\)ft ladder is leaning against a wall. The distance from the base of the ladder to the wall is \(4.5\) feet. How high on the wall can the ladder reach?

The Pythagorean Theorem says:

\begin{align*} a^2+b^2\amp=c^2\\ 4.5^2+b^2\amp=16.5^2\\ 20.25+b^2\amp=272.25\\ \end{align*}

Now we need to isolate \(b^2\) in order to solve for \(b\text{:}\)

\begin{align*} 20.25+b^2\subtractright{20.25}\amp=272.25\subtractright{20.25}\\ b^2\amp=252\\ \end{align*}

To remove the square, we use the square root property. Because this is a geometric situation we only need to use the principal root:

\begin{align*} b\amp=\sqrt{252}\\ \end{align*}

Now simplify this radical and then approximate it:

\begin{align*} b\amp=\sqrt{36\cdot 7}\\ b\amp=6\sqrt{7}\\ b\amp\approx 15.87 \end{align*}
a right angle with a diagonal line for the ladder; the base of the triangle is labeled a=4.5 ft;the height of the triangle is labeled b; the hypotenuse is labeled c=16.5 ft
Figure8.3.16Leaning Ladder

The ladder can reach about \(15.87\) feet high on the wall.