A car travels West for 79 miles before turning at an angle of 22° North of East. The car continues on this path for 63 miles and stops. How far is the car from its starting point? Round your answer to the nearest tenth.
To find the distance of the car from its starting point, we can use the concept of vectors.
First, let's break down the car's motion into two parts: the westward movement and the diagonal movement after turning.
1. Westward movement: The car travels west for 79 miles. Since it is moving in a straight line, we can represent this as a vector in the west direction. Let's call this vector W. The magnitude of W is 79 miles, and since it is going west, its direction is 180° or π radians.
2. Diagonal movement: After turning, the car travels at an angle of 22° north of east for 63 miles. To represent this motion as a vector, we need to break it down into its east and north components.
- Eastward component: To find the eastward component of the diagonal movement, we can use the cosine of the angle. The eastward component vector, E, has a magnitude of 63 miles multiplied by the cosine of 22°, and its direction is 0° or 0 radians.
- Northward component: To find the northward component of the diagonal movement, we can use the sine of the angle. The northward component vector, N, has a magnitude of 63 miles multiplied by the sine of 22°, and its direction is 90° or π/2 radians clockwise from the east direction.
Now, let's calculate the resultant vector R, which represents the total displacement of the car from its starting point.
R = W + E + N
To calculate the magnitude (distance) of the resultant vector, we can use the Pythagorean theorem:
|R| = √(|W|^2 + |E|^2 + |N|^2)
Finally, we can round the magnitude of the resultant vector R to the nearest tenth to get the answer.
Now let's calculate step by step:
1. Calculate the eastward component (E):
E = 63 miles * cos(22°) ≈ 57.145 miles
2. Calculate the northward component (N):
N = 63 miles * sin(22°) ≈ 23.031 miles
3. Calculate the magnitude of the resultant vector (R):
|R| = √(79^2 + 57.145^2 + 23.031^2) ≈ 100.6 miles
Therefore, the car is approximately 100.6 miles away from its starting point.