Question
A ladder that is 14 feet long is placed against a building. The bottom of the ladder is 6 feet from the base of the building.
A right triangle with side length 6 feet, hypotenuse 14 feet, and side h.
In feet, how high up the side of the building is the top of the ladder? Round to the nearest tenth of a foot.
A right triangle with side length 6 feet, hypotenuse 14 feet, and side h.
In feet, how high up the side of the building is the top of the ladder? Round to the nearest tenth of a foot.
Answers
GPT-4o mini
To find the height (h) up the side of the building that the ladder reaches, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is:
\[ c^2 = a^2 + b^2 \]
In this case:
- The length of the ladder (hypotenuse) \( c = 14 \) feet,
- The distance from the base of the building (one side of the triangle) \( a = 6 \) feet,
- The height up the building (the other side of the triangle) is \( b = h \).
We can rearrange the Pythagorean theorem to solve for \( h \):
\[ h^2 = c^2 - a^2 \]
Substituting the known values:
\[ h^2 = 14^2 - 6^2 \]
\[ h^2 = 196 - 36 \]
\[ h^2 = 160 \]
Now, we take the square root of both sides to find \( h \):
\[ h = \sqrt{160} \]
Calculating \( \sqrt{160} \):
\[ h \approx 12.649 \]
Rounding to the nearest tenth:
\[ h \approx 12.6 \]
Thus, the height up the side of the building that the top of the ladder reaches is approximately **12.6 feet**.
\[ c^2 = a^2 + b^2 \]
In this case:
- The length of the ladder (hypotenuse) \( c = 14 \) feet,
- The distance from the base of the building (one side of the triangle) \( a = 6 \) feet,
- The height up the building (the other side of the triangle) is \( b = h \).
We can rearrange the Pythagorean theorem to solve for \( h \):
\[ h^2 = c^2 - a^2 \]
Substituting the known values:
\[ h^2 = 14^2 - 6^2 \]
\[ h^2 = 196 - 36 \]
\[ h^2 = 160 \]
Now, we take the square root of both sides to find \( h \):
\[ h = \sqrt{160} \]
Calculating \( \sqrt{160} \):
\[ h \approx 12.649 \]
Rounding to the nearest tenth:
\[ h \approx 12.6 \]
Thus, the height up the side of the building that the top of the ladder reaches is approximately **12.6 feet**.