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 \( h \) at which the ladder touches the building, we can use the Pythagorean theorem. According to the theorem, the relationship in a right triangle is given by:
\[
a^2 + b^2 = c^2
\]
where:
- \( a \) is one leg of the triangle (the height \( h \)),
- \( b \) is the other leg of the triangle (the distance from the base of the building, which is 6 feet),
- \( c \) is the hypotenuse (the length of the ladder, which is 14 feet).
In this case, we have:
\[
h^2 + 6^2 = 14^2
\]
Calculating the squares:
\[
h^2 + 36 = 196
\]
Now, isolate \( h^2 \) by subtracting 36 from both sides:
\[
h^2 = 196 - 36
\]
\[
h^2 = 160
\]
Now, take the square root of both sides to find \( h \):
\[
h = \sqrt{160}
\]
Next, we can simplify \( \sqrt{160} \):
\[
h = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4\sqrt{10}
\]
Calculating \( \sqrt{10} \) (approximately 3.162):
\[
h \approx 4 \times 3.162 \approx 12.648
\]
Rounding to the nearest tenth:
\[
h \approx 12.6
\]
Thus, the height at which the top of the ladder touches the side of the building is approximately **12.6 feet**.
\[
a^2 + b^2 = c^2
\]
where:
- \( a \) is one leg of the triangle (the height \( h \)),
- \( b \) is the other leg of the triangle (the distance from the base of the building, which is 6 feet),
- \( c \) is the hypotenuse (the length of the ladder, which is 14 feet).
In this case, we have:
\[
h^2 + 6^2 = 14^2
\]
Calculating the squares:
\[
h^2 + 36 = 196
\]
Now, isolate \( h^2 \) by subtracting 36 from both sides:
\[
h^2 = 196 - 36
\]
\[
h^2 = 160
\]
Now, take the square root of both sides to find \( h \):
\[
h = \sqrt{160}
\]
Next, we can simplify \( \sqrt{160} \):
\[
h = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4\sqrt{10}
\]
Calculating \( \sqrt{10} \) (approximately 3.162):
\[
h \approx 4 \times 3.162 \approx 12.648
\]
Rounding to the nearest tenth:
\[
h \approx 12.6
\]
Thus, the height at which the top of the ladder touches the side of the building is approximately **12.6 feet**.