Question

if you place a 32 - foot ladder against the top of a 23 - foot building, how many feet will the bottom of the ladder be from the bottom of the building? round to the nearest tenth of a foot.

Explanation:

Step1: Identify the problem as a right - triangle problem

We can consider a right - triangle where the ladder is the hypotenuse ($c = 32$ feet) and the height of the building is one of the legs ($a = 23$ feet). We want to find the other leg ($b$). According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$.

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

We get $b=\sqrt{c^{2}-a^{2}}$. Substitute $c = 32$ and $a = 23$ into the formula: $b=\sqrt{32^{2}-23^{2}}=\sqrt{(32 + 23)(32 - 23)}$ (using the difference - of - squares formula $x^{2}-y^{2}=(x + y)(x - y)$). First, calculate $(32 + 23)(32 - 23)=(55)(9)=495$. Then $b=\sqrt{495}\approx22.2$ feet.

Answer:

$22.2$ feet