Question

kari is flying a kite. she releases 50 feet of string. what is the approximate difference in the height of the kite when the string makes a 25° angle with the ground and when the string makes a 45° angle with the ground? round to the nearest tenth.

Explanation:

Step1: Find height at 25°

The string is the hypotenuse (50 ft), height \( h_1 \) is opposite side. Use \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \), so \( h_1 = 50\sin(25^\circ) \). Calculate \( \sin(25^\circ)\approx0.4226 \), so \( h_1\approx50\times0.4226 = 21.13 \) feet.

Step2: Find height at 45°

For \( 45^\circ \), \( h_2 = 50\sin(45^\circ) \). \( \sin(45^\circ)=\frac{\sqrt{2}}{2}\approx0.7071 \), so \( h_2\approx50\times0.7071 = 35.355 \) feet.

Step3: Find the difference

Subtract \( h_1 \) from \( h_2 \): \( 35.355 - 21.13 = 14.225 \), round to nearest tenth is \( 14.2 \) feet. Wait, but let's recheck calculations. Wait, maybe I miscalculated \( \sin(25^\circ) \). Wait, \( \sin(25^\circ)\approx0.4226 \), 500.4226=21.13. \( \sin(45^\circ)\approx0.7071 \), 500.7071=35.355. Difference: 35.355 -21.13=14.225≈14.2. But the options have 14.2 feet. Wait, maybe my initial step had a miscalculation? Wait, no, let's recalculate \( \sin(25^\circ) \): using calculator, \( \sin(25^\circ)\approx0.422618 \), so 500.422618≈21.1309. \( \sin(45^\circ)\approx0.707107 \), 500.707107≈35.35535. Difference: 35.35535 -21.1309≈14.22445≈14.2. So the difference is approximately 14.2 feet.

Answer:

14.2 feet