An airplane is flying on a compass heading (bearing) of 170° at 460 mph. A wind is blowing with the bearing 200° at 80 mph. a. Find the component form of the velocity of the airplane. b. Find the actual ground speed and direction of the airplane.
An airplane is flying on a compass heading (bearing) of 170° at 460 mph. A wind is blowing with the bearing 200° at 80 mph. a. Find the component form of the velocity of the airplane. b. Find the actual ground speed and direction of the airplane.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter9: Quadratic Functions And Equations
Section9.7: Solving Systems Of Linear And Quadratic Equations
Problem 21PPS
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Find the error(s) and find the correct solution.
![### Problem:
An airplane is flying on a compass heading (bearing) of 170° at 460 mph. A wind is blowing with the bearing 200° at 80 mph.
#### Tasks:
a. Find the component form of the velocity of the airplane.
b. Find the actual ground speed and direction of the airplane.
### Solution:
#### Part (a):
To find the component form of the velocity of the airplane:
Given the airplane's speed and direction:
- Velocity \( v = 460 \) mph with a bearing of \( 170° \)
We convert this to component form using trigonometric functions:
\[ v = 460 \cdot \langle \cos 170°, \sin 170° \rangle \approx \langle -453.01, 79.88 \rangle \]
#### Part (b):
To find the ground speed and direction:
First, find the wind vector \( w \):
- Wind speed \( w = 80 \) mph with a bearing of \( 200° \)
We convert this to component form:
\[ w = 80 \cdot \langle \cos 200°, \sin 200° \rangle \approx \langle -75.18, -27.36 \rangle \]
Now, add the velocity vectors \( v \) and \( w \):
\[ \text{Velocity vector} = v + w = \langle -453.01, 79.88 \rangle + \langle -75.18, -27.36 \rangle \approx \langle -528.19, 52.52 \rangle \]
Calculate the actual ground speed:
\[ \text{Actual speed} = ||v + w|| = \sqrt{(-528.19)^2 + (52.52)^2} \approx 530.79 \text{ mph} \]
Calculate the actual direction \( \theta \):
\[ \theta = 180° + \tan^{-1} \left( \frac{-528.19}{52.52} \right) \approx 95.68° \]
### Summary:
- The component form of the velocity of the airplane is \(\langle -453.01, 79.88 \rangle\).
- The actual ground speed of the airplane is approximately 530.79 mph.
- The actual direction of the airplane is approximately 95.68](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F361784a8-d964-492d-9687-5d5c5e192aef%2F43a44f7a-dbb4-4d33-a153-915ba750d35c%2Fpzuj3vh_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem:
An airplane is flying on a compass heading (bearing) of 170° at 460 mph. A wind is blowing with the bearing 200° at 80 mph.
#### Tasks:
a. Find the component form of the velocity of the airplane.
b. Find the actual ground speed and direction of the airplane.
### Solution:
#### Part (a):
To find the component form of the velocity of the airplane:
Given the airplane's speed and direction:
- Velocity \( v = 460 \) mph with a bearing of \( 170° \)
We convert this to component form using trigonometric functions:
\[ v = 460 \cdot \langle \cos 170°, \sin 170° \rangle \approx \langle -453.01, 79.88 \rangle \]
#### Part (b):
To find the ground speed and direction:
First, find the wind vector \( w \):
- Wind speed \( w = 80 \) mph with a bearing of \( 200° \)
We convert this to component form:
\[ w = 80 \cdot \langle \cos 200°, \sin 200° \rangle \approx \langle -75.18, -27.36 \rangle \]
Now, add the velocity vectors \( v \) and \( w \):
\[ \text{Velocity vector} = v + w = \langle -453.01, 79.88 \rangle + \langle -75.18, -27.36 \rangle \approx \langle -528.19, 52.52 \rangle \]
Calculate the actual ground speed:
\[ \text{Actual speed} = ||v + w|| = \sqrt{(-528.19)^2 + (52.52)^2} \approx 530.79 \text{ mph} \]
Calculate the actual direction \( \theta \):
\[ \theta = 180° + \tan^{-1} \left( \frac{-528.19}{52.52} \right) \approx 95.68° \]
### Summary:
- The component form of the velocity of the airplane is \(\langle -453.01, 79.88 \rangle\).
- The actual ground speed of the airplane is approximately 530.79 mph.
- The actual direction of the airplane is approximately 95.68
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