Question

word problem involving the pythagorean theorem. a 9 ft ladder leans against the side of a house. the bottom of the ladder is 6 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 Pythagorean - theorem application

The ladder, the house, and the ground form a right - triangle. The length of the ladder is the hypotenuse $c = 9$ ft and the distance from the bottom of the ladder to the house is $a = 6$ ft. We want to find the height $b$ on the house where the top of the ladder reaches. The Pythagorean theorem is $a^{2}+b^{2}=c^{2}$.

Step2: Rearrange the formula to solve for $b$

We get $b=\sqrt{c^{2}-a^{2}}$. Substitute $c = 9$ and $a = 6$ into the formula: $b=\sqrt{9^{2}-6^{2}}=\sqrt{81 - 36}=\sqrt{45}$.

Step3: Calculate and round

$\sqrt{45}\approx6.7$ ft.

Answer:

$6.7$ ft