Question
Elton is using a ladder that is 20 feet long to reach a spot on a wall that is 16 feet above the ground. He places the ladder at the exact location he needs to reach. His employer uses safety measures that require the ladder to be between 72 degrees and 78 degrees angle at the base of the ladder. Using the inverse of tangent determine if Elton is within the allowed angle measures. What is the angle formed at the base of the ladder? Round your answer to the nearest whole degree
Answers
GPT-4o mini
To find the angle at the base of the ladder using the inverse tangent function, we can use the following relationship with the height of the wall and the length of the ladder:
1. **Identify the variables:**
- Height (\(h\)) = 16 feet (the height reached on the wall)
- Length of the ladder (\(L\)) = 20 feet
2. **Find the horizontal distance (\(d\)) from the base of the ladder to the wall using the Pythagorean theorem.**
According to the Pythagorean theorem:
\[
L^2 = h^2 + d^2
\]
Rearranging gives us:
\[
d^2 = L^2 - h^2
\]
Plugging in the values:
\[
d^2 = 20^2 - 16^2 = 400 - 256 = 144
\]
Therefore:
\[
d = \sqrt{144} = 12 \text{ feet}
\]
3. **Calculate the angle (\(\theta\)) using the inverse tangent function:**
\[
\theta = \tan^{-1}\left(\frac{h}{d}\right) = \tan^{-1}\left(\frac{16}{12}\right)
\]
Simplifying the fraction:
\[
\frac{16}{12} = \frac{4}{3}
\]
Now we calculate:
\[
\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ
\]
4. **Round to the nearest whole degree:**
\[
\theta \approx 53^\circ
\]
5. **Check if the angle is within the allowed measures (72 to 78 degrees):**
Since \(53^\circ\) is less than 72 degrees, it is not within the allowed angle measures.
**Conclusion:** The angle formed at the base of the ladder is approximately \(53^\circ\), which does not meet the safety requirement of being between 72 and 78 degrees.
1. **Identify the variables:**
- Height (\(h\)) = 16 feet (the height reached on the wall)
- Length of the ladder (\(L\)) = 20 feet
2. **Find the horizontal distance (\(d\)) from the base of the ladder to the wall using the Pythagorean theorem.**
According to the Pythagorean theorem:
\[
L^2 = h^2 + d^2
\]
Rearranging gives us:
\[
d^2 = L^2 - h^2
\]
Plugging in the values:
\[
d^2 = 20^2 - 16^2 = 400 - 256 = 144
\]
Therefore:
\[
d = \sqrt{144} = 12 \text{ feet}
\]
3. **Calculate the angle (\(\theta\)) using the inverse tangent function:**
\[
\theta = \tan^{-1}\left(\frac{h}{d}\right) = \tan^{-1}\left(\frac{16}{12}\right)
\]
Simplifying the fraction:
\[
\frac{16}{12} = \frac{4}{3}
\]
Now we calculate:
\[
\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ
\]
4. **Round to the nearest whole degree:**
\[
\theta \approx 53^\circ
\]
5. **Check if the angle is within the allowed measures (72 to 78 degrees):**
Since \(53^\circ\) is less than 72 degrees, it is not within the allowed angle measures.
**Conclusion:** The angle formed at the base of the ladder is approximately \(53^\circ\), which does not meet the safety requirement of being between 72 and 78 degrees.