Question
Tammy is an electrician. When using a ladder, her company requires the angle formed between the ground and the ladder to be within 3 degrees of 75 degrees so that she remains safe. If she is using a 20-foot ladder and places the ladder 5 feet from the wall, is this within safety protocol? What is the approximate angle formed between the ground and the ladder?
Answers
GPT-4o mini
To determine if Tammy's ladder placement is within the safe angle range specified by her company, we can use trigonometry. Specifically, we can use the tangent function since we know the height of the ladder and the distance from the wall.
Given:
- Length of the ladder (hypotenuse): \( L = 20 \) feet
- Distance from the wall (adjacent side): \( d = 5 \) feet
We use the Pythagorean theorem to find the height (opposite side) of the ladder when it is leaning against the wall. According to the theorem:
\[
h^2 + d^2 = L^2
\]
Where:
- \( h \) is the height.
- \( d \) is the distance from the wall (5 feet).
- \( L \) is the length of the ladder (20 feet).
Plugging in the known values, we have:
\[
h^2 + 5^2 = 20^2
\]
Calculating gives:
\[
h^2 + 25 = 400
\]
Now, solving for \( h^2 \):
\[
h^2 = 400 - 25 = 375
\]
Now, taking the square root:
\[
h = \sqrt{375} \approx 19.36 \text{ feet}
\]
Now that we have the height, we can calculate the angle \( \theta \) formed between the ground and the ladder using the cosine function:
\[
\cos(\theta) = \frac{d}{L} = \frac{5}{20} = 0.25
\]
To find \( \theta \), we take the arccosine:
\[
\theta \approx \cos^{-1}(0.25)
\]
Calculating this gives:
\[
\theta \approx 75.5^\circ
\]
Now we check if the angle \( \theta \) is within the required safety range of 72 degrees to 78 degrees (i.e., 75 degrees ± 3 degrees).
Since \( 75.5^\circ \) is within the allowable range, we conclude that Tammy's ladder placement is indeed within the safety protocol.
Thus, the approximate angle formed between the ground and the ladder is about **75.5 degrees**, which is within safety protocol.
Given:
- Length of the ladder (hypotenuse): \( L = 20 \) feet
- Distance from the wall (adjacent side): \( d = 5 \) feet
We use the Pythagorean theorem to find the height (opposite side) of the ladder when it is leaning against the wall. According to the theorem:
\[
h^2 + d^2 = L^2
\]
Where:
- \( h \) is the height.
- \( d \) is the distance from the wall (5 feet).
- \( L \) is the length of the ladder (20 feet).
Plugging in the known values, we have:
\[
h^2 + 5^2 = 20^2
\]
Calculating gives:
\[
h^2 + 25 = 400
\]
Now, solving for \( h^2 \):
\[
h^2 = 400 - 25 = 375
\]
Now, taking the square root:
\[
h = \sqrt{375} \approx 19.36 \text{ feet}
\]
Now that we have the height, we can calculate the angle \( \theta \) formed between the ground and the ladder using the cosine function:
\[
\cos(\theta) = \frac{d}{L} = \frac{5}{20} = 0.25
\]
To find \( \theta \), we take the arccosine:
\[
\theta \approx \cos^{-1}(0.25)
\]
Calculating this gives:
\[
\theta \approx 75.5^\circ
\]
Now we check if the angle \( \theta \) is within the required safety range of 72 degrees to 78 degrees (i.e., 75 degrees ± 3 degrees).
Since \( 75.5^\circ \) is within the allowable range, we conclude that Tammy's ladder placement is indeed within the safety protocol.
Thus, the approximate angle formed between the ground and the ladder is about **75.5 degrees**, which is within safety protocol.