Yahoo Answers: Answers and Comments for When are Ian and Ada closest together? [Mathematics]
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From wardadagreat
enCA
Tue, 20 Jul 2010 15:27:24 +0000
3
Yahoo Answers: Answers and Comments for When are Ian and Ada closest together? [Mathematics]
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https://ca.answers.yahoo.com/question/index?qid=20100720152724AASnq0G
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From SETH: Using Ada's house as the point of referenc...
https://ca.answers.yahoo.com/question/index?qid=20100720152724AASnq0G
https://ca.answers.yahoo.com/question/index?qid=20100720152724AASnq0G
Wed, 21 Jul 2010 15:12:23 +0000
Using Ada's house as the point of reference:
Ian is coming towards her house at 8 kph from 20 km away,
so the distance to Ada's house is 20  8t.
Ada is heading off at a right angle at 6 kph,
so the distance away from her house is 6t.
Since they are perpendicular angles from the house,
we can use the pythagorean theorem to calculate the distance:
Distance, d, is square root of (208t)^2 + (6t)^2
= sqrt (100t^2  320t + 400)
Now, it's tempting to think that they are at the minimum distance
when they start (t=0): 20km
or when they end (t=2.5): 15km
or when they are equidistant from Ada's house:
20  8t = 6t > t = 1.43 hrs > 12.1 km
But let's see if we can't use some calculus.
The minimum or maximum of an equation can be found
by setting the derivative equal to zero
(since the slope is zero).
We can ignore the square root since the maximum with it
will be at the same value for t as without it.
(if you do it out, you'll see it's the same...
the derivative is:
0.5 (200t320) / sqrt (100t^2 320t +400))
So ... derivative of 100t^2 + 320t + 400 equals 0
200t  320 = 0 > t = 1.6 hours
Ian is then 7.2 km away and Ada is 9.6 km away
For a distance, d, of 12 km.
We know that at t = 1.43 hours, d = 12.1 km
(from above conjecture at a solution).
What about at 1.7 hours?
Ian would be 6.4 km away
Ada would be 10.2 km away
They would be 12.04 km apart.
What about at 1.5 hours?
Ian would be 9 km away
Ada would be 8 km away
They would be 12.04 km apart.
So, I think we've proven your answer
though without doing the full derivative
or a second derivative
(with which we could prove whether
it was a minimum or maximum).
Depending upon the level of your class,
you may be asked upon to solve with
the full derivative equal to zero (min/max)
and the second derivative's value positive (minimum).
Ian and Ada are closest after 1 hour 16 minutes (1.6 hours)
when they are exactly 12 km apart.

From Susan: Speed X Time = Distance
Ian
8 X 2.5 = 20.0...
https://ca.answers.yahoo.com/question/index?qid=20100720152724AASnq0G
https://ca.answers.yahoo.com/question/index?qid=20100720152724AASnq0G
Tue, 20 Jul 2010 22:37:12 +0000
Speed X Time = Distance
Ian
8 X 2.5 = 20.0 km
Ada
6 x 2.5 = 15.0 km
They are closest when Ian reaches Ada's house he is now 15 km away. (At their homes, he is 20 km away)