7. The distance from the point directly under a hot air balloon to the point where the balloon is staked to the ground with a rope is 285 ft . The angle of elevation up the rope to the balloon is \( 48^{\circ} \). Find the height of the balloon.

To find the height of the balloon, we can use trigonometry. The scenario forms a right triangle where the distance from the point directly under the balloon to the stake is the adjacent side, and the height of the balloon is the opposite side. Using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] In this case, the height of the balloon (h) is the opposite side, and the distance (285 ft) is the adjacent side. Therefore: \[ \tan(48^{\circ}) = \frac{h}{285} \] Now, rearranging the equation gives us: \[ h = 285 \cdot \tan(48^{\circ}) \] Calculating this value: \[ h \approx 285 \cdot 1.1106 \approx 317.5 \text{ ft} \] So, the height of the balloon is approximately 317.5 feet.