Question

1. how far from the base of the house do you need to place a 15 - foot ladder so that it exactly reaches the top of a 12 - foot tall wall?

Explanation:

Step1: Identify the triangle type

This is a right - triangle problem, where the ladder is the hypotenuse ($c = 15$ feet), the height of the wall is one leg ($a = 12$ feet), and the distance from the base of the house to the ladder is the other leg ($b$) we need to find. We use the Pythagorean theorem: $a^{2}+b^{2}=c^{2}$.

Step2: Rearrange the formula to solve for b

We can rewrite the Pythagorean theorem as $b^{2}=c^{2}-a^{2}$. Substitute $c = 15$ and $a = 12$ into the formula: $b^{2}=15^{2}-12^{2}$. Calculate $15^{2}=225$ and $12^{2}=144$. Then $b^{2}=225 - 144=81$.

Step3: Solve for b

Take the square root of both sides: $b=\sqrt{81}=9$.

Answer:

9 feet