I'm learning about rational functions, and encountered this word problem:
A helicopter flies from Vancouver to Calgary a distance of 677km with
a tailwind. On the return trip the helicopter was 40km/h slower. The
total flying time for both flights was 6.5 hours. How fast was the
helicopter flying to Calgary? Round the answer to the nearest
hundredth.
Without looking, I tried:
$$\text{Total distance} / \text{Avg speed} = \text{Total time,}$$
$$\frac{1354}{s-20} = 6.5.$$
Which gives $(228.31 , 0)$ and looks more or less correct.
But the given solution is modeled as:
$$\text{Time with tailwind} + \text{Time with headwind} = \text{Total time}$$
$$\frac{677}{s} + \frac{677}{s-40} = 6.5.$$
Giving $(230.21,0)$.
I can tell at a glance that the given formulation is more precise and thus more likely to be correct. But as far as I can tell, what I came up with should work too. Where did I go wrong?
Best Answer
Your error was thinking that the average velocity was $s-20$. It is a common error.
Here’s a specific example:
Imagine you travel 60 km back and forth; if you travel the trip out at 60 kph, and the trip back at 40 kph, the average veloicity is not 50 kph, because your travel took two and a half hours, not two hours. So the average velocity turns out to be 48 kph.
When you divided by $s-20$, you asserted that because the trip out was done at $s$ kph, and the trip back at $s-40$ kph, then the average velocity was $\frac{1}{2}(s+s-40)=s-20$. It is not.
That’s your mistake, and why the solution divides the two trips, one at $s$ and one at $s-40$.