a tortoise crawling at a rate of 0.1 mph, the hare wants to rest 30 more minutes before chasing the turtle at 5 mph. how many feet must the hare run to catch the turtle?

during the 30 minutes, the tortoise travels

1/2 (.1) = .05 mi

so, if the hare runs for t hours,

5t = .05 + .1t
t = .0102 hours
at 5 mph, the hare runs 5*.0102 = .051 miles = 269.28 feet

check: the turtle has moved an additional .1*.0102 = 0.00102 miles, or .051 miles in all

The two distances are the same.

To calculate how many feet the hare must run to catch the turtle, we first need to determine the time it takes for the hare to start running.

Given that the tortoise crawls at a rate of 0.1 mph, we can convert this to feet per minute as follows:
0.1 mph = 0.1 * 5280 feet / 60 minutes = 0.88 feet per minute.

Now, let's calculate the time it takes for the hare to start running. We know that the hare wants to rest for an additional 30 minutes, so the total time before it starts chasing the turtle is 30 minutes.

Next, we can calculate the distance the tortoise crawls during this time:
Distance = Speed × Time
Distance = 0.88 feet per minute × 30 minutes = 26.4 feet.

Now, when the hare starts running, its speed is 5 mph, which we can also convert to feet per minute:
5 mph = 5 × 5280 feet / 60 minutes = 440 feet per minute.

To calculate the distance the hare must run to catch the turtle, we subtract the distance the tortoise crawled during the resting time from the distance the hare can cover in a minute:
Distance to run = 440 feet per minute × 30 minutes - 26.4 feet
Distance to run = 13200 feet - 26.4 feet
Distance to run = 13173.6 feet.

Therefore, the hare must run approximately 13173.6 feet to catch the turtle.

To determine the distance the hare must run to catch the turtle, we need to find out how long it will take for the hare to catch up with the tortoise.

First, let's convert the tortoise's speed from mph to feet per minute since we will be measuring the rest time of the hare in minutes.

1 mile = 5280 feet
1 hour = 60 minutes

Speed of the tortoise = 0.1 mph
Speed of the tortoise in feet per minute = 0.1 * 5280 / 60 = 88 feet per minute

Next, let's calculate the time it will take for the hare to catch up with the tortoise. The hare wants to rest for an additional 30 minutes before chasing the tortoise.

Time taken by the hare to catch up = Rest time + Time to catch up
Time to catch up = Distance / Speed

Rest time = 30 minutes
Speed of the hare = 5 mph
Speed of the hare in feet per minute = 5 * 5280 / 60 = 440 feet per minute

Time taken by the hare to catch up = 30 minutes + (Distance / 440 feet per minute)

Now, since the hare and the tortoise will cover the same distance when the hare catches up, we can create an equation:

Distance covered by the hare = Distance covered by the tortoise

(Distance / 440) = (Distance / 88)

Now, we can solve this equation to find the distance the hare must run to catch the tortoise.

(Distance / 440) = (Distance / 88)
Multiply both sides of the equation by 440 * 88
88 * Distance = 440 * Distance
440 * Distance - 88 * Distance = 0
(440 - 88) * Distance = 0
352 * Distance = 0
Distance = 0

The equation gives us Distance = 0, which means the hare and the tortoise will never meet. It seems like there might be another condition missing from the problem statement. Please check if there is any additional information provided or if any part of the problem has been left out.