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A 40 foot ladder is set against the side of a house so that it reaches up 24 feet. If Lily 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? (The answer is not
20ft.) Round to the nearest tenth of a foot.

A $40$ foot ladder is set against the side of a house so that it reaches up $24$ feet. If Lily 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? (The answer is not $20 \mathrm{ft}$ .) Round to the nearest tenth of a foot.

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Pythagorean theorem

Full solution

Q. A

40

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

24

feet. If Lily 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? (The answer is not

20 \mathrm{ft}

.) Round to the nearest tenth of a foot.

Calculate Original Distance: First, let's use the Pythagorean theorem to find the original distance from the house to the base of the ladder. $\newline$ We have a right triangle with the ladder as the hypotenuse $c = 40$ feet) and one leg representing the height the ladder reaches up the house $a = 24$ feet). We need to find the other leg $b$ , which is the original distance from the house to the base of the ladder. $\newline$ Using the Pythagorean theorem: $a^2 + b^2 = c^2$ . $\newline$ $24^2 + b^2 = 40^2$ .
Find New Distance: Now, let's calculate the original distance $b$ . $576 + b^2 = 1600$ . $b^2 = 1600 - 576$ . $b^2 = 1024$ . $b = \sqrt{1024}$ . $b = 32$ feet.So, the ladder was originally $32$ feet away from the house.
Calculate New Height: Next, Lily pulls the ladder extdollar{} $4$ extdollar{} feet farther from the house, so the new distance from the house to the base of the ladder is extdollar{} $32$ + $4$ = $36$ extdollar{} feet.
Calculate New Height: Next, Lily pulls the ladder $4$ feet farther from the house, so the new distance from the house to the base of the ladder is $32 + 4 = 36$ feet.Now we need to find the new height the ladder reaches up the house (let's call it $a_{\text{new}}$ ) with the new base distance of $36$ feet. $\newline$ Using the Pythagorean theorem again: $a_{\text{new}}^2 + 36^2 = 40^2$ . $\newline$ $a_{\text{new}}^2 + 1296 = 1600$ . $\newline$ $a_{\text{new}}^2 = 1600 - 1296$ . $\newline$ $a_{\text{new}}^2 = 304$ . $\newline$ $a_{\text{new}} = \sqrt{304}$ . $\newline$ $a_{\text{new}} \approx 17.4$ feet.

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