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Math Tutoring

<< Previous: Constraints and Viable Solutions

Back to Math 1: Unit 4

Next: Linear Inequalities >>

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.

<< Previous: Constraints and Viable Solutions

Back to Math 1: Unit 4

Next: Linear Inequalities >>

Keywords: Wind, headwind, tailwind, current, systems of linear equations, applications of linear equations, word problems
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