Plane A is 40 mi south and 100 mi east of Plane B. Plane A is flying 2 miles west for every mile it flies north, while Plane B is flying 3 mi east for every mile it flies south.
a. Where do their paths cross?
b. Which plane must fly farther?
c. What ratio of the speed of Plane B to the speed of Plane A would
produce a midair collision?
See:
http://www.jiskha.com/display.cgi?id=1254150016
Follow the formula that it goes by .
To solve this problem, let's break it down step-by-step:
Step 1: Draw a diagram
Draw a diagram representing the location of Plane A and Plane B. Place Plane B at the origin (0, 0) and Plane A at the coordinates (100, -40) since it is 40 miles south and 100 miles east of Plane B.
Step 2: Determine the direction coefficients
From the given information, we know that Plane A is flying 2 miles west for every mile it flies north, while Plane B is flying 3 miles east for every mile it flies south. This means the direction vectors for Plane A and Plane B are (-2, 1) and (3, -1), respectively.
Step 3: Find the intersection point
To find where their paths cross, we need to solve the following system of equations:
100A + 40B = 0 (equation for Plane A's path)
-2A + B = 0 (equation for Plane B's path)
Solving this system of equations, we get:
A = 8
B = 16
So the paths of Plane A and Plane B cross at the coordinates (8, 16).
Step 4: Calculate the distances traveled
To determine which plane flies farther, we need to calculate the distances traveled by each plane.
The distance flown by Plane A is the distance between the starting point and the intersection point:
Distance_A = sqrt((8 - 100)^2 + (16 - (-40))^2) ≈ 114.6 miles
The distance flown by Plane B is the distance between the starting point (0, 0) and the intersection point (8, 16):
Distance_B = sqrt(8^2 + 16^2) ≈ 18.4 miles
Therefore, Plane A flies farther than Plane B.
Step 5: Calculate the ratio of speeds for a mid-air collision
To determine the ratio of the speed of Plane B to the speed of Plane A that would produce a mid-air collision, we calculate the ratio of the distances flown:
Ratio = Distance_B / Distance_A
Ratio = 18.4 / 114.6
Ratio ≈ 0.1605
So for a mid-air collision, the speed of Plane B would need to be roughly 0.1605 times the speed of Plane A.
To solve this problem, we can use algebraic equations and concepts of distance, speed, and ratios. Let's break down each question step by step:
a. Where do their paths cross?
To determine where their paths cross, we need to find the coordinates of the intersection point. Let's assign variables to the distances flown by each plane:
- For Plane A: x miles north and (100 - 2x) miles west
- For Plane B: y miles south and (3y) miles east
Since their paths cross, the coordinates of the intersection point will be the same for both planes. We can set up equations based on this:
For the latitude (north-south):
x = y
For the longitude (east-west):
100 - 2x = 3y
Now we can solve this system of equations:
Substitute x = y into the second equation:
100 - 2y = 3y
Combine like terms:
100 = 5y
Divide both sides by 5:
y = 20
Substitute y = 20 back into the first equation:
x = 20
Therefore, the paths of Plane A and Plane B cross at coordinates (20, 20).
b. Which plane must fly farther?
To determine which plane must fly farther, we need to compare the total distance flown by both planes. For Plane A, the distance flown is x + (100 - 2x) = 100 - x. For Plane B, the distance flown is y + (3y) = 4y. Since we calculated that y = 20, the distance flown by Plane B is 4(20) = 80 miles. The distance flown by Plane A can be found by substituting x = 20: 100 - x = 100 - 20 = 80 miles.
Therefore, both planes must fly the same distance, which is 80 miles.
c. What ratio of the speed of Plane B to the speed of Plane A would produce a midair collision?
To determine the ratio of the speed of Plane B to the speed of Plane A for a midair collision, we need to consider the rate of change in their distances flown.
Plane A flies 2 miles west for every 1 mile north, giving it a speed ratio of 2:1 (west:north). Plane B flies 3 miles east for every 1 mile south, giving it a speed ratio of 3:1 (east:south).
To have a midair collision, their rates must be equal. So, we need to find the ratio of the speeds of Plane B to Plane A.
The speed ratio of Plane B to Plane A is (3:1) / (2:1) = 3/2.
Therefore, a ratio of 3:2 (speed of Plane B to speed of Plane A) would produce a midair collision.