Alright, let's solve the problem step-by-step to find the difference in the height of the kite when the string makes a 25° angle with the ground and when it makes a 45° angle with the ground.
1. Given Information:
- The length of the string is 50 feet.
- We have two angles with the ground: 25° and 45°.
2. Understanding the Heights:
To find the height of the kite at different angles, we will use the sine function, which relates the angle in a right triangle to the ratio of the opposite side (height of the kite) to the hypotenuse (length of the string).
3. Calculating Height at 25°:
- The height ([tex]\(h_{25}\)[/tex]) can be calculated using the sine function: [tex]\( \sin(25^\circ) = \frac{h_{25}}{50} \)[/tex].
- Solving for [tex]\(h_{25}\)[/tex]: [tex]\( h_{25} = 50 \cdot \sin(25^\circ) \)[/tex].
- Using the sine of 25°, we find that [tex]\( h_{25} \approx 21.13 \)[/tex] feet.
4. Calculating Height at 45°:
- The height ([tex]\(h_{45}\)[/tex]) can similarly be calculated: [tex]\( \sin(45^\circ) = \frac{h_{45}}{50} \)[/tex].
- Solving for [tex]\(h_{45}\)[/tex]: [tex]\( h_{45} = 50 \cdot \sin(45^\circ) \)[/tex].
- Using the sine of 45°, we find that [tex]\( h_{45} \approx 35.36 \)[/tex] feet.
5. Calculating the Difference in Heights:
- The difference in height is calculated by subtracting the height at 25° from the height at 45°: [tex]\( \Delta h = |h_{45} - h_{25}| \)[/tex].
- Plugging the values in: [tex]\( \Delta h = |35.36 - 21.13| \)[/tex].
- Thus, the difference in height [tex]\( \Delta h \approx 14.23 \)[/tex] feet.
6. Rounding to the Nearest Tenth:
- Rounding 14.23 to the nearest tenth, we get 14.2 feet.
Therefore, the approximate difference in the height of the kite when the string makes a 25° angle with the ground and when it makes a 45° angle with the ground is 14.2 feet.
