Question

if you place a 23 - foot ladder against the top of a building and the bottom of the ladder is 11 feet from the bottom of the building, how tall is the building? round to the nearest tenth of a foot.

Explanation:

Step1: Identify the problem as a right - triangle problem

We can consider the ladder as the hypotenuse of a right - triangle, the distance from the bottom of the ladder to the bottom of the building as one leg, and the height of the building as the other leg. Let the height of the building be $h$, the length of the ladder $c = 23$ feet and the base $a = 11$ feet. By the Pythagorean theorem $a^{2}+h^{2}=c^{2}$.

Step2: Rearrange the Pythagorean theorem to solve for $h$

We get $h=\sqrt{c^{2}-a^{2}}$. Substitute $c = 23$ and $a = 11$ into the formula: $h=\sqrt{23^{2}-11^{2}}=\sqrt{(23 + 11)(23 - 11)}=\sqrt{34\times12}=\sqrt{408}$.

Step3: Calculate the value of $h$

$\sqrt{408}\approx20.2$

Answer:

$20.2$ feet