Question
A person in a hot air balloon is looking down with an angle of 36° at their car 75 feet away (straight line distance - diagonal). How high is the balloon? Round your answer to the nearest tenth of a foot.
Answers
Answered by
GPT-4o mini
To find the height of the hot air balloon, we can use trigonometry. We are given the angle of depression and the straight-line distance to the car (75 feet).
Since the person in the balloon is looking down at an angle of 36°, we can use the sine function, which relates the opposite side (the height of the balloon) to the hypotenuse (the straight-line distance).
Let's denote the height of the balloon as \( h \).
Using the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
In this case:
\[
\sin(36^\circ) = \frac{h}{75}
\]
To find \( h \), we can rearrange the equation:
\[
h = 75 \cdot \sin(36^\circ)
\]
Next, we calculate \( \sin(36^\circ) \):
\[
\sin(36^\circ) \approx 0.5878
\]
Now we substitute this value back into the equation:
\[
h = 75 \cdot 0.5878 \approx 44.085
\]
Rounding this to the nearest tenth of a foot gives:
\[
h \approx 44.1 \text{ feet}
\]
Thus, the height of the balloon is approximately **44.1 feet**.
Since the person in the balloon is looking down at an angle of 36°, we can use the sine function, which relates the opposite side (the height of the balloon) to the hypotenuse (the straight-line distance).
Let's denote the height of the balloon as \( h \).
Using the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
In this case:
\[
\sin(36^\circ) = \frac{h}{75}
\]
To find \( h \), we can rearrange the equation:
\[
h = 75 \cdot \sin(36^\circ)
\]
Next, we calculate \( \sin(36^\circ) \):
\[
\sin(36^\circ) \approx 0.5878
\]
Now we substitute this value back into the equation:
\[
h = 75 \cdot 0.5878 \approx 44.085
\]
Rounding this to the nearest tenth of a foot gives:
\[
h \approx 44.1 \text{ feet}
\]
Thus, the height of the balloon is approximately **44.1 feet**.