Question

the base of a 14 - foot ladder is 2 feet from a building. if the ladder reaches the flat roof, how tall is the building? the height of the building is □ ft. (simplify your answer. type an exact answer, using radicals as needed.) the height of the building is approximately □ ft. (round to the nearest tenth as needed.)

Explanation:

Step1: Apply Pythagorean theorem

Let the height of the building be $h$. The Pythagorean theorem is $a^{2}+b^{2}=c^{2}$, where $c$ is the length of the ladder (hypotenuse), $a$ is the distance from the base of the ladder to the building, and $b$ is the height of the building. Here, $c = 14$ and $a=2$. So, $2^{2}+h^{2}=14^{2}$.

Step2: Solve for $h$

First, expand the equation: $4 + h^{2}=196$. Then, subtract 4 from both sides: $h^{2}=196 - 4=192$. Take the square - root of both sides: $h=\sqrt{192}$. Simplify $\sqrt{192}=\sqrt{64\times3}=8\sqrt{3}$.

Step3: Find the approximate value

Calculate the approximate value of $8\sqrt{3}$. $\sqrt{3}\approx1.732$, so $8\sqrt{3}\approx8\times1.732 = 13.9$.

Answer:

The height of the building is $8\sqrt{3}$ ft.
The height of the building is approximately $13.9$ ft.