tried various ways of solving

Question

On a certain day, Donnie observes the wind is blowing at 4 miles per hour. A bird nesting near Donnie's house flies three quarters of a mile down the road (in the direction of the wind), turns around, and returns exactly 6 minutes later. What is the airspeed of the bird?

Answer 1

Jacquelyn,

 

I can see how this is a bit of a challenging problem! It took me a minute to organize it. I will show you how to do it generically, that is, with wind speed w, distance d, and speed v. Once we have the solution there, we can simply plug in those values and calculate the answer.

 

First off, we recall that v=d/t, where v is speed, d is distance and t is time.

We can rearrange that to t=d/v by simply swapping the t with the v.

 

This is the big step! Now I will write out the total time of flight given all the variables:

 

d/(v+w) +d/(v-w) = t

 

This is the sum of the times for the flight down the road and then back up the road. One of the legs of the trip is with the wind and one is against - it doesn't matter which is which in this problem. It will also be important later to ensure the units are consistent.

 

Now we will multiply both sides of the equation by (v+w)(v-w) to remove the denominators. Note that in doing this, we stipulate that v≠w or -w since that would be multiplying by zero. Physically, -w makes no sense anyway. If the bird's speed is w, he can make no headway against the wind so that one won't happen either.

 

The equation now becomes:

d(v-w) + d(v+w) = t(v-w)(v+w)

 

Distributing both sides:

2dv = t(v²-w²) 

 

since the +dw and -dw terms cancel on the left and the +vw and -vw terms cancel on the right.

 

Subtract 2dv from both sides:

 

0=tv²-2dv+tw²

 

You could solve this symbolically now using the quadratic formula but let's plug in the givens from the problem and go from there. Remember, we are solving for v. Note that although the round trip time is given as 6 minutes, that has to be converted to hours (0.1) because we are using miles per hour as the speed units.

 

0=0.1v²-2(0.75)v+(0.1)(4)²

0=0.1v²-1.5v+1.6

 

Let's multiply both sides by 10 to make all the coefficients integers

 

0=v²-15v+16

 

Factoring this:

 

0=(v-16)(v+1)

 

So:

 

v=16 or -1

 

We can disregard the -1 because its not a real solution to this problem, physically.

 

Therefore, the bird's airspeed is 16 mph.