Question
Colton is flying a kite, holding his hands a distance of 3 feet above the ground and letting all the kite’s string play out. He measures the angle of elevation from his hand to the kite to be 32degrees
∘
. If the string from the kite to his hand is 90 feet long, how many feet is the kite above the ground? Round your answer to the nearest hundredth of a foot if necessary.
∘
. If the string from the kite to his hand is 90 feet long, how many feet is the kite above the ground? Round your answer to the nearest hundredth of a foot if necessary.
Answers
GPT-4o mini
To find the height of the kite above the ground, we can use trigonometry. The relevant trigonometric function in this case is the sine function, which relates the angle of elevation, the height of the kite above his hands, and the length of the string.
1. Let \( h \) be the height of the kite above Colton's hands.
2. We have a right triangle where:
- The opposite side is \( h \) (the height of the kite above his hands).
- The hypotenuse is the length of the string, which is 90 feet.
- The angle of elevation is \( 32^\circ \).
Using the sine function:
\[
\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Thus, we can write:
\[
\sin(32^\circ) = \frac{h}{90}
\]
3. Now we can solve for \( h \):
\[
h = 90 \cdot \sin(32^\circ)
\]
4. Using a calculator, we find:
\[
\sin(32^\circ) \approx 0.5299
\]
5. Substituting this value in:
\[
h = 90 \cdot 0.5299 \approx 47.691
\]
Thus, \( h \approx 47.69 \) feet.
6. Finally, to find the total height of the kite above the ground, we add the height of Colton's hands (which is 3 feet):
\[
\text{Total height} = h + \text{height of hands} = 47.69 + 3 \approx 50.69
\]
Therefore, the kite is approximately **50.69 feet** above the ground.
1. Let \( h \) be the height of the kite above Colton's hands.
2. We have a right triangle where:
- The opposite side is \( h \) (the height of the kite above his hands).
- The hypotenuse is the length of the string, which is 90 feet.
- The angle of elevation is \( 32^\circ \).
Using the sine function:
\[
\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Thus, we can write:
\[
\sin(32^\circ) = \frac{h}{90}
\]
3. Now we can solve for \( h \):
\[
h = 90 \cdot \sin(32^\circ)
\]
4. Using a calculator, we find:
\[
\sin(32^\circ) \approx 0.5299
\]
5. Substituting this value in:
\[
h = 90 \cdot 0.5299 \approx 47.691
\]
Thus, \( h \approx 47.69 \) feet.
6. Finally, to find the total height of the kite above the ground, we add the height of Colton's hands (which is 3 feet):
\[
\text{Total height} = h + \text{height of hands} = 47.69 + 3 \approx 50.69
\]
Therefore, the kite is approximately **50.69 feet** above the ground.