Question

the base of a 10 - foot ladder is 5 feet from a building. if the ladder reaches the flat roof, how tall is the building? the height of the building is \\( \square \\) ft. (simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Apply Pythagorean theorem

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 building (one - leg), and \(b\) is the height of the building (the other leg). Given \(c = 10\) (length of the ladder) and \(a=5\) (distance from the building). We need to find \(b\).
Substitute into the formula: \(b^{2}=c^{2}-a^{2}\)

Step2: Calculate \(b^{2}\)

\(b^{2}=10^{2}-5^{2}\)
\(b^{2}=100 - 25\)
\(b^{2}=75\)

Step3: Solve for \(b\)

\(b=\sqrt{75}\)
Simplify \(\sqrt{75}=\sqrt{25\times3}\)
Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 25\), \(b = 3\)), we get \(b = 5\sqrt{3}\)

Answer:

\(5\sqrt{3}\)