Question

if you place a 38-foot ladder against the top of a 32-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 type

This is a right triangle problem where the ladder is the hypotenuse (\(c = 38\) feet), the building height is one leg (\(a = 32\) feet), and we need to find the other leg (\(b\)) which is the distance from the bottom of the ladder to the bottom of the building. We use the Pythagorean theorem: \(a^{2}+b^{2}=c^{2}\).

Step2: Rearrange the formula to solve for \(b\)

We can rearrange the Pythagorean theorem to solve for \(b\): \(b=\sqrt{c^{2}-a^{2}}\).

Step3: Substitute the values

Substitute \(c = 38\) and \(a = 32\) into the formula: \(b=\sqrt{38^{2}-32^{2}}\). First, calculate \(38^{2}=1444\) and \(32^{2}=1024\). Then, \(38^{2}-32^{2}=1444 - 1024=420\). So, \(b=\sqrt{420}\).

Step4: Calculate the square root and round

\(\sqrt{420}\approx20.5\) (rounded to the nearest tenth).

Answer:

The bottom of the ladder will be approximately \(\boldsymbol{20.5}\) feet from the bottom of the building.