a plane flies 1.5 hours at 110 mph on a bearing of 10°. it then turns and flies 8.5 hours at the same speed on a bearing of 100°. How far is the plane from its starting point?
A bearing is a direction you look.
A HEADING is a direction you sail or fly.
anyway:
10 degrees heading is 10 degrees east of north
100 degrees heading is 10 degrees south of east
so
North distance = 1.5*110 cos 10 - 8.5*110 sin 10
East distance = 1.5*110 sin 10 + 8.5*110 cos 10
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factor 110 out of all of that and add it back at the end
n = 110 (.0012)
e = 110 (8.63)
d^2 = n^2+e^2 = 110 sqrt(.0012^2+8.63^2)
=110(8,63)
= 949 miles
Please add parentheses
To find the distance the plane is from its starting point, we can use the concept of vector addition.
Let's break down the plane's journey into two segments:
Segment 1:
Time = 1.5 hours
Speed = 110 mph
Bearing = 10°
Segment 2:
Time = 8.5 hours
Speed = 110 mph
Bearing = 100°
First, let's calculate the distance traveled in each segment using the formula:
Distance = Speed x Time
Segment 1 distance = 110 mph x 1.5 hours = 165 miles
Segment 2 distance = 110 mph x 8.5 hours = 935 miles
To find the total distance, we need to calculate the resultant vector by adding the two individual vectors.
To add vectors, we can break them down into their horizontal (East-West) and vertical (North-South) components.
For Segment 1:
Horizontal component = Distance x cos(Bearing)
Vertical component = Distance x sin(Bearing)
Horizontal component of Segment 1 = 165 miles x cos(10°) ≈ 164.04 miles
Vertical component of Segment 1 = 165 miles x sin(10°) ≈ 28.62 miles
For Segment 2:
Horizontal component = Distance x cos(Bearing)
Vertical component = Distance x sin(Bearing)
Horizontal component of Segment 2 = 935 miles x cos(100°) ≈ -852.64 miles
Vertical component of Segment 2 = 935 miles x sin(100°) ≈ 119.75 miles
Now, let's calculate the total East-West component and North-South component by adding the respective components from each segment:
Total East-West component = Horizontal component of Segment 1 + Horizontal component of Segment 2
Total East-West component ≈ 164.04 miles + (-852.64 miles) ≈ -688.6 miles
Total North-South component = Vertical component of Segment 1 + Vertical component of Segment 2
Total North-South component ≈ 28.62 miles + 119.75 miles ≈ 148.37 miles
Finally, we can use the Pythagorean theorem to find the total distance from the starting point:
Total distance = √[(Total East-West component)^2 + (Total North-South component)^2]
Total distance ≈ √[(-688.6 miles)^2 + (148.37 miles)^2]
Total distance ≈ √[474205.96 miles^2 + 22006.2769 miles^2]
Total distance ≈ √[496212.2369 miles^2]
Total distance ≈ 704.60 miles
Therefore, the plane is approximately 704.60 miles from its starting point.
To find the distance the plane is from its starting point, we can use the concept of vector addition. Essentially, we will break down the plane's movement into its horizontal and vertical components, and then find the total displacement.
Here's how we can solve it step by step:
Step 1: Determine the horizontal and vertical components of the plane's movement for each leg of the flight.
For the first leg:
- Time: 1.5 hours
- Speed: 110 mph
- Bearing: 10°
We can find the horizontal and vertical components using trigonometry. The horizontal component (dx) is the distance covered in the east-west direction, and the vertical component (dy) is the distance covered in the north-south direction.
dx = speed * cos(bearing) = 110 mph * cos(10°)
dy = speed * sin(bearing) = 110 mph * sin(10°)
For the second leg:
- Time: 8.5 hours
- Speed: 110 mph
- Bearing: 100°
Similarly, we calculate the horizontal and vertical components:
dx = speed * cos(bearing) = 110 mph * cos(100°)
dy = speed * sin(bearing) = 110 mph * sin(100°)
Step 2: Calculate the total horizontal and vertical displacement.
To find the total horizontal displacement, we add up the horizontal components of both legs:
total_dx = dx_leg1 + dx_leg2
To find the total vertical displacement, we add up the vertical components:
total_dy = dy_leg1 + dy_leg2
Step 3: Calculate the distance (d) from the starting point using the Pythagorean theorem.
d = sqrt(total_dx^2 + total_dy^2)
Now, let's calculate the values:
dx_leg1 = 110 mph * cos(10°)
dy_leg1 = 110 mph * sin(10°)
dx_leg2 = 110 mph * cos(100°)
dy_leg2 = 110 mph * sin(100°)
total_dx = dx_leg1 + dx_leg2
total_dy = dy_leg1 + dy_leg2
d = sqrt(total_dx^2 + total_dy^2)
Calculating these values should give you the distance the plane is from its starting point.