Applications of Linear Equations: Wind/Current Problems
Standard
A.CED.3 Represent constraints by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Concept Check: Wind Speeds
An airplane experiences headwind and tailwind.
- Headwind is against the flow. This means that the plane is being pushed back and that its speed will be slower.
- Tailwind is with the flow. This means that the plane is being pushed forward and that its speed will be faster.
We can model such situations with equations:
Headwind (against the flow)
air speed - wind speed = ground speed
Tailwind (with the flow)
air speed + wind speed = ground speed
Air speed represents the speed that the plane is going as if there were no forces acting upon it.
Wind speed is self-explanatory. It's the speed of the wind.
Ground speed represents the overall speed, or the observable speed.
Example
With a tailwind, a bird flew at a ground speed of 4 mi/h. Flying the same path against the same wind, the bird travels at a ground speed of 2 mi/h. What is the bird's air speed? What is the wind speed?
List your givens
This is just really good practice. You can also assign variables to unknowns that you have.
- Let x = air speed
- Let y = wind speed
- Ground speed is 4 mi/h with tailwind
- Ground speed is 2 mi/h with headwind
Here's our equations
Headwind
x - y = 2
Tailwind
x + y = 4
We have a system of equations and we have several ways to find the solution. Which would be the best method to use? Graphing, substitution, or elimination?
Elimination
Elimination looks like the easiest option to do. Here's the steps:
- Add the equations together in order to cancel out y.
- Divide by 2 to get x.
Solve for y
Remember that once you have x, you can take this value and plug it into any of the equations. Here, we have chosen the second equation. Here's the steps:
- Plug in 3 for x.
- Subtract 3 on both sides of the equation.
Interpret results
We listed in the beginning what x and y are.
- x is the air speed. So this bird, on its own, is flying at 3 mi/h.
- y is the wind speed. The wind is blowing at 1 mi/h. Quite the gentle breeze.
Example
You row upstream at a speed of 4 mi/h. You travel the same distance downstream at a speed of 7 mi/h. What would be your rowing speed in still water? What is the speed of the current?
This is very similar to the wind speed problem! We're just working in terms of currents.
So our rowing speed will be affected by the current. If the current is with us we go faster (downstream), and if it is against us we go slower (upstream).
Against the current
rowing speed - current speed = upstream speed
With the current
rowing speed + current speed = downstream speed
Givens
- Let x = rowing speed.
- Let y = current speed.
- Upstream speed is 4 mi/h
- Downstream speed is 7 mi/h
Equations
x - y = 4
x + y = 7
Elimination
Interpret Results
- x is rowing speed. So if the water was still and there was no current, the boat would be moving at 5.5 mi/h.
- y is the current speed. The river is flowing at a speed of 1.5 mi/h.