I need help setting up this equation

Question

a small plane, flying into a head wind, flew 500 miles in 7 hours. Flying with the wind, the plane traveled 1150 miles in 4 hours. Find the rate of the plane is calm air and the rate of the wind. Round to the nearest whole number

Answer 1

This will require a system of two equations, because there are two variables, airspeed and wind speed. Lets call them A and W, respectively. The first equation will come from the case with the headwind, so A and W are acting in opposite directions.

 

A-W=500miles/7hours, which is about 71.4 mph when traveling upwind.

 

The second equation is for downwind travel, so A and W act in the same direction:

 

A+W=1150miles/4hours, which is 287.5 mph downwind.

 

Next, pick an equation (doesn't matter which) and solve for one of the variables, A or W (again, it doesn't matter which). Use the resulting equation to substitute for that variable in the other equation.

 

For instance, I will solve the downwind equation for wind speed:

 

W=(287.5 mph) - A

 

Now, I substitute the right side of this equation for W in the upwind equation:

 

A - (287.5 - A) = 71.4

 

Solve this for A and you have your airspeed (don't forget to distribute the minus sign to both terms inside the parentheses). You can use this value in either of the original equations to find wind speed.

 

Note: solving for A initially, instead of W, will require the last step to find airspeed, whereas solving for W does not. (If airspeed is all we are interested in.)