Question: A person flying a kite has released 176m of string. The string makes an angle of 27 degrees with the ground. How high is the kite? How far away is the kite horizontally? Give your answer in the nearest meter.
1. **State the problem.**
We have a right triangle where the string is the hypotenuse with length $176$ m, and the angle with the ground is $27^\circ$.
We need to find:
- the height of the kite, and
- the horizontal distance from the person to the kite.
2. **Write the formulas.**
For a right triangle:
$$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$$
$$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$$
Here, $\theta=27^\circ$ and the hypotenuse is $176$.
3. **Find the height.**
The height is the side opposite the angle, so:
$$\sin(27^\circ)=\frac{h}{176}$$
Multiply both sides by $176$:
$$h=176\sin(27^\circ)$$
Now evaluate:
$$h\approx 176(0.4540)$$
$$h\approx 79.9$$
Rounded to the nearest meter:
$$h\approx 80\text{ m}$$
4. **Find the horizontal distance.**
The horizontal distance is the side adjacent to the angle, so:
$$\cos(27^\circ)=\frac{x}{176}$$
Multiply both sides by $176$:
$$x=176\cos(27^\circ)$$
Now evaluate:
$$x\approx 176(0.8910)$$
$$x\approx 156.8$$
Rounded to the nearest meter:
$$x\approx 157\text{ m}$$
5. **Final answer.**
The kite is about $80$ m high and about $157$ m away horizontally.