Answer:
2 minutes.
Step-by-step explanation:
Let t be the time taken in competition and the distance covered by Achilles be x miles.
We have been given that when Achilles reaches the finish line, the Turtle is 2/5 of a mile behind him. This means that distance covered by turtle will be: [tex]x-\frac{2}{5}[/tex].
Achilles runs at a speed of 13 mph while the Turtle moves at a speed of 1 mph.
Since we know that [tex]\text{Time}=\frac{\text{Distance}}{\text{Speed}}[/tex], so we can set two equations for time taken by Achilles and turtle as:
[tex]t=\frac{x}{13}...(1)[/tex]
[tex]t=\frac{(x-\frac{2}{5})}{1}...(2)[/tex]
From equation (1) we will get,
[tex]x=13t[/tex]
Upon substituting this value in equation (2) we will get,
[tex]t=\frac{13t-\frac{2}{5}}{1}[/tex]
Let us have a common denominator.
[tex]t=\frac{\frac{65t-2}{5}}{1}[/tex]
[tex]t=\frac{65t-2}{5}[/tex]
Upon multiplying both sides of our equation by 5 we will get,
[tex]5t=5\times \frac{65t-2}{5}[/tex]
[tex]5t=65t-2[/tex]
[tex]5t-65t=65t-65t-2[/tex]
[tex]-60t=-2[/tex]
[tex]t=\frac{-2}{-60}[/tex]
[tex]t=\frac{1}{30}[/tex]
So, the competition took 1/30 hour, let us convert our given time in minutes.
[tex]\frac{1}{30}\text{ hour}\times \frac{60\text{ minutes}}{\text{hour}}[/tex]
[tex]\frac{1}{30}\times 60\text{ minutes}[/tex]
[tex]2\text{ minutes}[/tex]
Therefore, the competition took 2 minutes.
