Solve the Walking Puzzle: Find 2.5 Miles in 30 Minutes

[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

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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

Homework Equations

The Attempt at a Solution

I thought about writing :

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 :)

 

[Dick]

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.

 

[JPC]

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 ?

 

[Dick]

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

 

[JPC]

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

 

[Feldoh]

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?

 

[JPC]

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