To find the height of the hot air balloon, we can use the tangent trigonometric ratio. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side.
In this scenario:
- The angle of elevation from the antenna to the balloon is 63°.
- The distance from the antenna to the library (the adjacent side) is 24 miles.
- We want to find the height of the balloon above the antenna (the opposite side).
Using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] where \(\theta = 63°\).
Plugging in the values: \[ \tan(63°) = \frac{\text{height of the balloon}}{24} \]
We can rearrange this to solve for the height of the balloon: \[ \text{height of the balloon} = 24 \cdot \tan(63°) \]
Now, we can calculate: \[ \text{height of the balloon} \approx 24 \cdot 1.9626 \quad (\text{using a calculator for } \tan(63°)) \] \[ \text{height of the balloon} \approx 47.1 \text{ miles} \]
Thus, the height of the hot air balloon is approximately 47.1 miles.
Trigonometric ratio used: Tangent (tan)