Question

a 9-ft ladder leans against the side of a house. the bottom of the ladder is 5 ft from the side of the house. how high is the top of the ladder from the ground? if necessary, round your answer to the nearest tenth.

Explanation:

Step1: Identify the problem type

This is a right triangle problem where the ladder is the hypotenuse, the distance from the house is one leg, and the height on the house is the other leg. We use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\), \(b\) are the legs. Let \(a\) be the height (unknown), \(b = 5\) ft, and \(c = 9\) ft.

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

We need to find \(a\), so rearrange the Pythagorean theorem: \(a^2 = c^2 - b^2\). Substitute \(c = 9\) and \(b = 5\): \(a^2 = 9^2 - 5^2\). Calculate \(9^2 = 81\) and \(5^2 = 25\), so \(a^2 = 81 - 25 = 56\).

Step3: Solve for \(a\)

Take the square root of both sides: \(a = \sqrt{56}\). Calculate \(\sqrt{56} \approx 7.5\) (rounded to the nearest tenth).

Answer:

\(7.5\)