Question

a 40 foot ladder is set against the side of a house so that it reaches up 24 feet. if aiden grabs the ladder at its base and pulls it 5 feet farther from the house, how far up the side of the house will the ladder reach now? (the answer is not 19 ft.) round to the nearest tenth of a foot.

Explanation:

Step1: Find the initial distance from the house

We can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 40\) (ladder length) and \(b=24\) (height on the house). Let \(a\) be the initial distance from the house. So \(a^{2}+24^{2}=40^{2}\). Then \(a^{2}=40^{2}-24^{2}\). Calculate \(40^{2}=1600\) and \(24^{2} = 576\). So \(a^{2}=1600 - 576=1024\), then \(a=\sqrt{1024} = 32\) feet.

Step2: Find the new distance from the house

After pulling the ladder 5 feet farther, the new distance from the house is \(32 + 5=37\) feet.

Step3: Find the new height on the house

Let the new height be \(h\). Using the Pythagorean theorem again with \(c = 40\) and \(a = 37\). So \(h^{2}+37^{2}=40^{2}\). Then \(h^{2}=40^{2}-37^{2}\). Calculate \(40^{2}=1600\) and \(37^{2}=1369\). So \(h^{2}=1600 - 1369 = 231\). Then \(h=\sqrt{231}\approx15.2\) feet.

Answer:

\(15.2\) feet