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A $35$ foot ladder is set against the side of a house so that it reaches up $21$ feet. If Elijah grabs the ladder at its base and pulls it $4$ feet farther from the house, how far up the side of the house will the ladder reach now?

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Pythagorean theorem

Full solution

Q. A

35

foot ladder is set against the side of a house so that it reaches up

21

feet. If Elijah grabs the ladder at its base and pulls it

4

feet farther from the house, how far up the side of the house will the ladder reach now?

Identify setup: Identify the initial setup of the ladder and the house. $\newline$ The ladder forms a right triangle with the house and the ground. The hypotenuse (ladder) is $35$ feet, and the height (side of the house reached by the ladder) is $21$ feet.
Calculate initial distance: Calculate the initial distance from the base of the ladder to the house using the Pythagorean theorem. $\newline$ Let $x$ be the initial distance from the base of the ladder to the house. $\newline$ $x^2 + 21^2 = 35^2$ $\newline$ $x^2 + 441 = 1225$ $\newline$ $x^2 = 1225 - 441$ $\newline$ $x^2 = 784$ $\newline$ $x = \sqrt{784}$ $\newline$ $x = 28$ feet.
Update distance: Update the distance from the base of the ladder to the house after pulling it $4$ feet farther. $\newline$ New distance from the house = $28$ feet + $4$ feet = $32$ feet.
Calculate new height: Calculate the new height up the side of the house the ladder reaches using the Pythagorean theorem again. $\newline$ Let $y$ be the new height. $\newline$ $32^2 + y^2 = 35^2$ $\newline$ $1024 + y^2 = 1225$ $\newline$ $y^2 = 1225 - 1024$ $\newline$ $y^2 = 201$ $\newline$ $y = \sqrt{201}$ $\newline$ $y = 14.177$ feet.

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