To solve this problem, 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 over the adjacent side.
Given:
- Angle \( \theta = 63^\circ \)
- Distance from the radio antenna to the library (adjacent side) = 24 miles
- Height of the balloon (opposite side) = ?
Trigonometric Ratio Used:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] For our problem: \[ \tan(63^\circ) = \frac{\text{height of the balloon}}{24 \text{ miles}} \]
Setting Up the Equation:
Let \( h \) be the height of the balloon. So we have: \[ \tan(63^\circ) = \frac{h}{24} \]
Rearranging the equation to solve for \( h \):
\[ h = 24 \times \tan(63^\circ) \]
Calculating \( \tan(63^\circ) \):
Using a calculator, we find: \[ \tan(63^\circ) \approx 1.9626 \]
Now substituting back into the equation:
\[ h = 24 \times 1.9626 \] \[ h \approx 47.1 \text{ miles} \]
Final Answer:
The height of the balloon is approximately 47.1 miles.