Question

a 50 foot ladder is set against the side of a house so that it reaches up 30 feet. if alonso 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 25 ft.) round to the nearest tenth of a foot.

Explanation:

Step1: Find the original distance of the ladder base from the house

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 50\) (ladder length) and \(b = 30\) (height on the house). Let the original base - distance be \(a\).

$$a=\sqrt{c^{2}-b^{2}}=\sqrt{50^{2}-30^{2}}=\sqrt{(50 + 30)(50 - 30)}=\sqrt{80\times20}=\sqrt{1600}=40$$

Step2: Find the new base - distance

The new base - distance \(a_{new}=40 + 5=45\)

Step3: Find the new height on the house

Using the Pythagorean theorem again, with \(c = 50\) and \(a_{new}=45\). Let the new height be \(b_{new}\)

$$b_{new}=\sqrt{c^{2}-a_{new}^{2}}=\sqrt{50^{2}-45^{2}}=\sqrt{(50 + 45)(50 - 45)}=\sqrt{95\times5}=\sqrt{475}\approx21.8$$

Answer:

The ladder will reach approximately \(21.8\) feet up the side of the house.