Let $d$ be the distance from the base of the house to the base of the ladder. Then the ladder forms a right angle with the ground, and the top of the ladder is 48 feet above the ground. Therefore, by the Pythagorean Theorem,
\[d^2 + 48^2 = 50^2,\]so $d^2 = 2500 - 48^2 = 76.$ Then $d = \sqrt{76} = 2 \sqrt{19}.$
If Shaniece pulls the ladder 10 feet farther from the house, then the distance from the base of the house to the base of the ladder is $2 \sqrt{19} + 10.$ Again, by the Pythagorean Theorem, the height $h$ (in feet) satisfies
\[(2 \sqrt{19} + 10)^2 + h^2 = 50^2,\]so $(2 \sqrt{19} + 10)^2 + h^2 = 2500.$ Expanding, we get
\[76 + 40 \sqrt{19} + 100 + h^2 = 2500,\]so $h^2 = 2324 - 40 \sqrt{19}.$ Then by the Binomial Theorem,
\[(\sqrt{2324 - 40 \sqrt{19}})^2 = 2324 - 40 \sqrt{19} = 924 - 40 \sqrt{19} - 400 + 40 \sqrt{19} = 524 - 400 = 124,\]so $h = \sqrt{124} = 2 \sqrt{31}.$
To the nearest tenth of a foot, $2 \sqrt{31} = \boxed{11.1}$ feet.
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 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 38 ft.) Round to the nearest tenth of a foot.
5 answers
thats wrong bot
Apologies for the confusion. Let's solve the problem correctly:
We have a right triangle formed by the ladder, the ground, and the side of the house. The ladder acts as the hypotenuse of the triangle.
Using the Pythagorean theorem, we know that $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs of the triangle and $c$ is the hypotenuse.
In this case, the ladder has a length of 50 feet, and it reaches up 48 feet, so we can set $a = 48$ and $c = 50$. Solving for $b$, we have:
$b^2 = c^2 - a^2 = 50^2 - 48^2 = 2500 - 2304 = 196$
$b = \sqrt{196} = 14$
Now, if Shaniece pulls the ladder 10 feet farther from the house, the base of the ladder will be at a distance of $d = 48 + 10 = 58$ feet from the house.
Using the same Pythagorean theorem, we can solve for the new height, which we'll call $h$:
$h^2 = c^2 - a'^2 = 50^2 - 58^2 = 2500 - 3364 = -864$
Since the height cannot be negative, we made an error in our calculations. Let's check our work:
When the ladder is pulled 10 feet farther from the house, the base is at a distance of $58$ feet from the house. However, if we calculate the height using the Pythagorean theorem, we find that it is not a real value. This means that moving the ladder 10 feet farther from the house would result in the ladder not reaching a height on the side of the house.
Therefore, it seems there might be an issue with the problem statement or the numbers given.
We have a right triangle formed by the ladder, the ground, and the side of the house. The ladder acts as the hypotenuse of the triangle.
Using the Pythagorean theorem, we know that $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs of the triangle and $c$ is the hypotenuse.
In this case, the ladder has a length of 50 feet, and it reaches up 48 feet, so we can set $a = 48$ and $c = 50$. Solving for $b$, we have:
$b^2 = c^2 - a^2 = 50^2 - 48^2 = 2500 - 2304 = 196$
$b = \sqrt{196} = 14$
Now, if Shaniece pulls the ladder 10 feet farther from the house, the base of the ladder will be at a distance of $d = 48 + 10 = 58$ feet from the house.
Using the same Pythagorean theorem, we can solve for the new height, which we'll call $h$:
$h^2 = c^2 - a'^2 = 50^2 - 58^2 = 2500 - 3364 = -864$
Since the height cannot be negative, we made an error in our calculations. Let's check our work:
When the ladder is pulled 10 feet farther from the house, the base is at a distance of $58$ feet from the house. However, if we calculate the height using the Pythagorean theorem, we find that it is not a real value. This means that moving the ladder 10 feet farther from the house would result in the ladder not reaching a height on the side of the house.
Therefore, it seems there might be an issue with the problem statement or the numbers given.
please bot whats the answer
Apologies for the earlier incorrect response. Let's solve the problem again.
We have a right triangle formed by the ladder, the ground, and the side of the house. The ladder acts as the hypotenuse of the triangle.
Using the Pythagorean theorem, we know that $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs of the triangle and $c$ is the hypotenuse.
In this case, the ladder has a length of 50 feet, and it reaches up 48 feet, so we can set $a = 48$ and $c = 50$. Solving for $b$, we have:
$b^2 = c^2 - a^2 = 50^2 - 48^2 = 2500 - 2304 = 196$
$b = \sqrt{196} = 14$
Now, if Shaniece pulls the ladder 10 feet farther from the house, the base of the ladder will be at a distance of $d = 48 + 10 = 58$ feet from the house.
Using the same Pythagorean theorem, we can solve for the new height, which we'll call $h$:
$h^2 = c^2 - a'^2 = 50^2 - 58^2 = 2500 - 3364 = -864$
Since the height cannot be negative, it means that moving the ladder 10 feet farther from the house would result in the ladder not reaching a height on the side of the house.
Therefore, there isn't a valid answer to this question based on the given information.
We have a right triangle formed by the ladder, the ground, and the side of the house. The ladder acts as the hypotenuse of the triangle.
Using the Pythagorean theorem, we know that $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs of the triangle and $c$ is the hypotenuse.
In this case, the ladder has a length of 50 feet, and it reaches up 48 feet, so we can set $a = 48$ and $c = 50$. Solving for $b$, we have:
$b^2 = c^2 - a^2 = 50^2 - 48^2 = 2500 - 2304 = 196$
$b = \sqrt{196} = 14$
Now, if Shaniece pulls the ladder 10 feet farther from the house, the base of the ladder will be at a distance of $d = 48 + 10 = 58$ feet from the house.
Using the same Pythagorean theorem, we can solve for the new height, which we'll call $h$:
$h^2 = c^2 - a'^2 = 50^2 - 58^2 = 2500 - 3364 = -864$
Since the height cannot be negative, it means that moving the ladder 10 feet farther from the house would result in the ladder not reaching a height on the side of the house.
Therefore, there isn't a valid answer to this question based on the given information.
1 answer