Answer
The distance traveled by the hare would be approximately $269.28$ feet.
Work Step by Step
We need to know that, $distance=speed\times time$.
The turtle has already crawled for $0.5$ hours at a speed of $0.1$ miles per hour when the hare starts to move.
Therefore, the distance that the turtle has already traveled is,
$0.1\times 0.5=0.05$ miles
When the hare starts to move, the turtle continues to crawl at a speed of $0.1$ miles per hour, whereas the hare moves at a speed of $5$ miles per hour.
After $x$ hours, the turtle would have traveled another $0.1x$ miles, whereas the heir would have traveled $5x$ miles.
To find the number of hours after which the hare will meet the turtle, we set up the equation: $0.1x+0.05=5x$
We multiply 100 by both sides of the equation to clear the equation from decimals.
$100(0.1x+0.005)=5x\times 100$
$10x+5=500x$
We subtract $10x$ from both sides of the equation.
$10x+5-10x=500x-10x$
$5=490x$
Dividing both sides of the equation by 490.
$\frac{5}{490}=\frac{490x}{490}$
$x\approx 0.0102$ hours
Therefore, the total distance the hare would have traveled is
$5x=5\times 0.0102$
$= 0.51$miles
$=269.28$ feet
