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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.

Question $\newline$ Watch Video $\newline$ Show Examples $\newline$ 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. Question

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Watch Video

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Show Examples

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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

20 \mathrm{ft}

.) Round to the nearest tenth of a foot.

Identify Problem: question_prompt: How far up the side of the house will the ladder reach now after being pulled $4$ feet farther from the house?
Original Triangle: Original ladder position forms a right triangle with the house and ground. The ladder is the hypotenuse, $40$ feet. The height reached is one leg, $24$ feet. We need to find the original distance from the house, which is the other leg.
Pythagorean Theorem: Use Pythagorean Theorem: $a^2 + b^2 = c^2$ , where $c$ is the ladder length, $a$ is the height reached, and $b$ is the original distance from the house. Plug in the known values: $24^2 + b^2 = 40^2$ .
Calculate Original Distance: Calculate the original distance from the house: $24^2 + b^2 = 40^2$ , so $576 + b^2 = 1600$ . Subtract $576$ from both sides to get $b^2 = 1600 - 576$ .
Find Original Distance: Continue calculation: $b^2 = 1600 - 576$ , so $b^2 = 1024$ . Take the square root of both sides to find $b$ : $b = \sqrt{1024}$ .
Calculate New Distance: Find the original distance from the house: $b = \sqrt{1024}$ , so $b = 32$ feet.
New Triangle: Lily pulls the ladder $4$ feet farther from the house, so the new distance from the house is $32 + 4 = 36$ feet.
Calculate New Height: Now we have a new right triangle with the ladder still as the hypotenuse ( $40$ feet) and the new distance from the house ( $36$ feet). We need to find the new height reached using the Pythagorean Theorem: $a^2 + 36^2 = 40^2$ .
Calculate New Height: Calculate the new height reached: $a^2 + 36^2 = 40^2$ , so $a^2 + 1296 = 1600$ . Subtract $1296$ from both sides to get $a^2 = 1600 - 1296$ .
Final Height: Continue calculation: $a^2 = 1600 - 1296$ , so $a^2 = 304$ . Take the square root of both sides to find $a$ : $a = \sqrt{304}$ .
Final Height: Continue calculation: $a^2 = 1600 - 1296$ , so $a^2 = 304$ . Take the square root of both sides to find $a$ : $a = \sqrt{304}$ . Find the new height reached: $a = \sqrt{304}$ , so $a \approx 17.4$ feet. Round to the nearest tenth of a foot.

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