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### Averaging 30 Miles Per Hour

```
Date: 03/17/97 at 17:58:41
From: by way of Sarah Seastone
Subject: How to Average 30 MPH

You go on a car trip. For half the distance, you travel at 15 mph.
What speed should you travel for the rest of your trip such that you
average 30 mph?

Hint:  Don't assume the obvious!

Michelle
```

```
Date: 03/18/97 at 17:21:24
From: Doctor Anthony
Subject: Re: How to Average 30 MPH

Suppose half the distance is x. Time = dist/speed = x/15

Then for the second half of the journey, Time = x/v (v = speed on
second half)

Total time = x/15 + x/v   Total distance = 2x

Average speed for whole journey =  Total dist/Total time
= 2x/(x/15 + x/v)
=  2/(1/15 + 1/v)

and this must equal 30.   So 2/(1/15 + 1/v) = 30
1/15 = 1/15 + 1/v
0 = 1/v  so  v = infinity

This shows that you would require an infinite speed on the second half
of the journey to get an average speed of 30 mph.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 03/18/97 at 14:16:03
From: Doctor Trent
Subject: Re: Word Problem

You can't do it, the trip you ask for is impossible.

Let's answer this without algebra by looking at an example. Assume
that you drove that first half in exactly one hour. That means you
went 15 miles. The second half of the trip is then another 15 miles,
for a total of 30 miles. To average 30 miles per hour, you have to
finish the whole 30 miles in one hour, but you've already used up an
hour, so you have no time left to travel the other 15 miles.

This example would work with any time that you put in; one hour was
just the simplest.

You're right, don't assume!

-Doctor Trent,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
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