To determine the length of the kite string given the height and the angle it makes with the ground, you can follow these steps:
1. Identify the known values:
- The height of the kite above the ground (opposite side of the right triangle) is 32 meters.
- The angle between the kite string and the ground is 39°.
2. Use trigonometry:
- The relationship between the height (opposite side) and the kite string (hypotenuse) in a right triangle is given by the sine (sin) function:
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
where [tex]\(\theta\)[/tex] is the angle.
3. Set up the equation:
- Rearrange the formula to solve for the hypotenuse (kite string length):
[tex]\[
\text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)}
\][/tex]
4. Convert the angle to radians:
- Trigonometric functions in many calculators and mathematical contexts use radians. Convert 39° to radians:
[tex]\[
\theta = 39^\circ \approx 0.681 \text{ radians}
\][/tex]
5. Calculate the hypotenuse:
- Using the sine function:
[tex]\[
\sin(0.681) \approx 0.629
\][/tex]
[tex]\[
\text{hypotenuse} = \frac{32 \text{ meters}}{0.629} \approx 50.85 \text{ meters}
\][/tex]
6. Round the result:
- Round 50.85 meters to the nearest meter:
[tex]\[
\text{hypotenuse} \approx 51 \text{ meters}
\][/tex]
Therefore, the length of the kite string, to the nearest meter, is approximately 51 meters.
