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Original Question
A 50 foot ladder is set against the side of a house so that it reaches up 48 feet. If Shaniece grabs the ladder at its base and...Question
A 30 foot ladder is set against the side of a house so that it reaches up 24 feet. If Alexander grabs the ladder at its base and pulls it 10 feet farther from the house, how far up the side of the house will the ladder reach now? (The answer is not 14 ft.) Round to the nearest tenth of a foot.
Answers
GPT 3.5
When Alexander pulls the ladder 10 feet farther from the house, he creates a right triangle with the ladder as the hypotenuse, the vertical distance the ladder reaches up the house as one leg, and the 10 feet he pulled the ladder away from the house as the other leg.
Using the Pythagorean theorem, we can calculate the new vertical distance the ladder reaches up the house:
\( c^2 = a^2 + b^2 \)
Where:
- c = length of the ladder = 30 feet
- a = original vertical distance = 24 feet
- b = 10 feet
\( c^2 = 24^2 + 10^2 \)
\( c^2 = 576 + 100 = 676 \)
\( c = \sqrt{676} \)
\( c = 26 \) feet (rounded to the nearest foot)
Therefore, when Alexander pulls the ladder 10 feet farther from the house, the ladder reaches up 26 feet along the side of the house.
Using the Pythagorean theorem, we can calculate the new vertical distance the ladder reaches up the house:
\( c^2 = a^2 + b^2 \)
Where:
- c = length of the ladder = 30 feet
- a = original vertical distance = 24 feet
- b = 10 feet
\( c^2 = 24^2 + 10^2 \)
\( c^2 = 576 + 100 = 676 \)
\( c = \sqrt{676} \)
\( c = 26 \) feet (rounded to the nearest foot)
Therefore, when Alexander pulls the ladder 10 feet farther from the house, the ladder reaches up 26 feet along the side of the house.
This is wrong
GPT 3.5
Apologies for the mistake. Let's correct it.
After Alexander pulls the ladder 10 feet farther from the house, we can create a right triangle with one leg as the vertical distance the ladder reaches up the house and the other leg as the 10 feet he pulled the ladder away from the house. The ladder remains the hypotenuse of this triangle.
Given that the ladder is 30 feet, the ladder initially reaches up 24 feet, and Alexander pulls the ladder 10 feet farther from the house, we can calculate the new vertical distance the ladder reaches up the house using the Pythagorean theorem:
\(Total Ladder Length^2 = Vertical Distance^2 + Distance Pulled Away^2\)
\(30^2 = 24^2 + 10^2\)
\(900 = 576 + 100\)
\(900 = 676 + 100\)
\(900 = 776\)
\(Vertical Distance^2 = 900 - 100\)
\(Vertical Distance = \sqrt{800}\)
\(Vertical Distance = 28.28\) feet (rounded to the nearest hundredth)
Therefore, when Alexander pulls the ladder 10 feet farther from the house, the ladder reaches up to approximately 28.3 feet along the side of the house.
After Alexander pulls the ladder 10 feet farther from the house, we can create a right triangle with one leg as the vertical distance the ladder reaches up the house and the other leg as the 10 feet he pulled the ladder away from the house. The ladder remains the hypotenuse of this triangle.
Given that the ladder is 30 feet, the ladder initially reaches up 24 feet, and Alexander pulls the ladder 10 feet farther from the house, we can calculate the new vertical distance the ladder reaches up the house using the Pythagorean theorem:
\(Total Ladder Length^2 = Vertical Distance^2 + Distance Pulled Away^2\)
\(30^2 = 24^2 + 10^2\)
\(900 = 576 + 100\)
\(900 = 676 + 100\)
\(900 = 776\)
\(Vertical Distance^2 = 900 - 100\)
\(Vertical Distance = \sqrt{800}\)
\(Vertical Distance = 28.28\) feet (rounded to the nearest hundredth)
Therefore, when Alexander pulls the ladder 10 feet farther from the house, the ladder reaches up to approximately 28.3 feet along the side of the house.