To solve this problem, we need to break down the boat's path into its northward and eastward components using trigonometry. The boat travels 24 miles at a bearing of N 43° E. This means that the path forms a right triangle where the 24 miles is the hypotenuse, and the angle from the north to the path is 43°.
Here are the steps to solve the problem:
1. Identify the given data:
- Distance traveled (hypotenuse), [tex]\( d = 24 \)[/tex] miles
- Bearing angle, [tex]\( \theta = 43^\circ \)[/tex]
2. Calculate the northward component:
- The northward component can be found by using the cosine function, which relates the adjacent side of the triangle to the hypotenuse:
[tex]\[
\text{Distance North} = d \cos(\theta)
\][/tex]
Substituting the given values,
[tex]\[
\text{Distance North} = 24 \cos(43^\circ)
\][/tex]
3. Calculate the eastward component:
- The eastward component can be found by using the sine function, which relates the opposite side of the triangle to the hypotenuse:
[tex]\[
\text{Distance East} = d \sin(\theta)
\][/tex]
Substituting the given values,
[tex]\[
\text{Distance East} = 24 \sin(43^\circ)
\][/tex]
4. Determine the final answers:
- Using the cosine of 43°, which gives the northward component, and rounding to the nearest tenth:
[tex]\[
\text{Distance North} \approx 17.6 \text{ miles}
\][/tex]
- Using the sine of 43°, which gives the eastward component, and rounding to the nearest tenth:
[tex]\[
\text{Distance East} \approx 16.4 \text{ miles}
\][/tex]
Therefore, the boat has traveled approximately 17.6 miles north and 16.4 miles east from the harbor.
The correct answer is:
C) 17.6 miles north and 16.4 miles east
