Question

a 14 - foot ladder is leaning against a building, with the base of the ladder 5 feet from the building. how high up on the building will the top of the ladder reach? the ladder will reach ( 3sqrt{19} ) ft high up on the building. (simplify your answer. type an exact answer, using radicals as needed.) the ladder will reach ( square ) ft high up on the building. (round to the nearest tenth.)

Explanation:

Step1: Use the Pythagorean theorem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (length of the ladder, \(c = 14\) feet) and \(a\) is the base (distance from the building, \(a=5\) feet), and \(b\) is the height on the building. We need to solve for \(b\), so \(b=\sqrt{c^{2}-a^{2}}\).
Substitute \(c = 14\) and \(a = 5\) into the formula: \(b=\sqrt{14^{2}-5^{2}}=\sqrt{196 - 25}=\sqrt{171}\). Simplify \(\sqrt{171}=\sqrt{9\times19}=3\sqrt{19}\) (exact value).

Step2: Calculate the approximate value

We know that \(\sqrt{19}\approx4.359\). Then \(3\sqrt{19}\approx3\times4.359 = 13.077\approx13.1\) (rounded to the nearest tenth).

Answer:

The ladder will reach \(3\sqrt{19}\) ft high up on the building (exact value) and \(13.1\) ft high up on the building (rounded to the nearest tenth).