Question
What is the length of the ladder leaning against a house at a 60-degree angle, with the bottom 7 feet away from the foundation?
Answers
GPT-4o mini
To find the length of the ladder leaning against the house, we can use trigonometric principles. In this setup, we have a right triangle where:
- The angle between the ladder and the ground is \(60^\circ\).
- The distance from the base of the ladder to the house (the horizontal leg) is 7 feet.
- The length of the ladder is the hypotenuse.
We can use the cosine function, which relates the adjacent side (the horizontal distance) to the hypotenuse. The formula is:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Substituting in our values:
\[
\cos(60^\circ) = \frac{7}{L}
\]
where \(L\) is the length of the ladder. The cosine of \(60^\circ\) is \(0.5\):
\[
0.5 = \frac{7}{L}
\]
To solve for \(L\), we can rearrange the equation:
\[
L = \frac{7}{0.5}
\]
Calculating this gives:
\[
L = 14 \text{ feet}
\]
So, the length of the ladder is **14 feet**.
- The angle between the ladder and the ground is \(60^\circ\).
- The distance from the base of the ladder to the house (the horizontal leg) is 7 feet.
- The length of the ladder is the hypotenuse.
We can use the cosine function, which relates the adjacent side (the horizontal distance) to the hypotenuse. The formula is:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Substituting in our values:
\[
\cos(60^\circ) = \frac{7}{L}
\]
where \(L\) is the length of the ladder. The cosine of \(60^\circ\) is \(0.5\):
\[
0.5 = \frac{7}{L}
\]
To solve for \(L\), we can rearrange the equation:
\[
L = \frac{7}{0.5}
\]
Calculating this gives:
\[
L = 14 \text{ feet}
\]
So, the length of the ladder is **14 feet**.