# Solve the Walking Puzzle: Find 2.5 Miles in 30 Minutes

• JPC
In summary, the conversation discusses how to prove the existence of a time interval of 30 minutes during which someone has walked exactly 2.5 miles, given that they walked 5 miles in one hour. The use of the intermediate value theorem is suggested to show this proof, and the conversation also touches on the importance of clearly verifying all the hypotheses.
JPC

## Homework Statement

Hello
Its a pretty tricky question ! Help me find the answer so i can finally sleep (i only get the answer when i hand in the question sheets in 3 weeks) :D

->>>>

You walk for 1 hour, and do 5 miles during that hour

Question : show that there exist a time interval of length 30minutes during which you have walked 2.5 miles

## The Attempt at a Solution

dD = Vi * dt
where D : distance
t = time
(Vi = dP / dt)

And integrating dD between 0 and 1hour. and saying that the whole is equal to 5 miles
and then separate the integral into 3 ones where one is from 0 to 30 minutes

but it doesn't lead to much conclusions

Any ideas would be appreciated, thank you :)

Call M(t) the distance you walk between time t and time t+30 minutes. Think about M(0) and M(30). Is it possible they are both less than 2.5? Is it possible they are both greater than 2.5? M(t) is continuous as t goes from 0 to 30 minutes.

sorry for the late reply, i had an exam, 1 oral test, 2 big homeworks in between :D

Hum, yes we have M(0) + M(30) = 5
one is necessarily bigger than the other if they are not equal. But then, if one is bigger than the other, how do you justify that there is a 30minutes time interval during which you have walked exactly 2.5 miles ?

Because M(t) is a continuous function for t in [0,30]. Think about using the intermediate value theorem.

oh, i understand now, thank you :)

if M(0) = a
M(30) = b

we have a+b = 5
we suppose 'a' different than 'b'

M(t) continuous on the interval [0, 30]
so there exists a 't' belonging to R where M(t) = 2.5

You still haven't proven anything. What you're saying is equivalent to me saying:
"if M(0) = a
M(30) = b

we have a+b = 5
we suppose 'a' different than 'b'

M(t) continuous on the interval [0, 30]
so there exists a dog named ralph, somewhere"

Why is this the case?

oh, this is not what i would write, first of all because i am in France ( i would write it in french), and finally because i just quickly resumed it here.

I will of course clearly show i have verified all the hypothesis of the intermediate value theorem. And i better do, my maths professor is more rigorous than my calculator :D

## 1. How do I solve the walking puzzle?

To solve the walking puzzle and find 2.5 miles in 30 minutes, you will need to walk at an average speed of 5 miles per hour. This means you will need to walk at a pace of 12 minutes per mile.

## 2. How can I improve my walking speed?

To improve your walking speed, you can try incorporating interval training into your routine. This involves alternating between short bursts of fast walking and slower recovery periods. Additionally, make sure to maintain good posture and take longer strides while walking.

## 3. What are some tips for completing the puzzle in less than 30 minutes?

To complete the walking puzzle in less than 30 minutes, you will need to increase your walking speed. This can be achieved by practicing regularly, staying hydrated, and wearing comfortable shoes. You can also try walking on flat terrain and avoiding distractions while walking.

## 4. How can I track my walking distance and time?

There are many apps and fitness trackers available that can help you track your walking distance and time. You can also use a pedometer or simply use a map to measure your route and a timer to track your time.

## 5. Is walking a good form of exercise?

Yes, walking is a great form of exercise as it can improve cardiovascular health, strengthen muscles, and help with weight loss. It is also a low-impact activity, making it suitable for people of all ages and fitness levels.

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