Help with a calculus problem about optimization
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Help with a calculus problem about optimization

[From: ] [author: ] [Date: 11-11-29] [Hit: ]
Distance divided by speed gives time.So the above adds up two times -- the time by water plus the time by land.To optimize this, you need to find the derivative and solve it with respect to x and set that equal to 0 (which means either a minimum or maximum) and then find x from that.I assume that is about where your class is at, right now.......

So now you should be able to write out T(x), which needs to be in units of time, as:

1.    T(x) = [ √(2^2+x^2) ] / (3 miles/hr) + ( 6 - x ) / (4 miles/hr)
2.    T(x) = (√[x^2+4])/3 + 6/4 - x/4

If you look at it, the distances are in the numerators and the speeds are in the denominators. Distance divided by speed gives time. So the above adds up two times -- the time by water plus the time by land.

To "optimize" this, you need to find the derivative and solve it with respect to x and set that equal to 0 (which means either a minimum or maximum) and then find x from that. I assume that is about where your class is at, right now. Anyway, so take the derivative:

3.    d( T(x) ) = d( (√[x^2+4])/3 + 6/4 - x/4 )
4.    d( T(x) ) = d( (√[x^2+4])/3 ) + d( 6/4 ) - d( x/4 )
5.    d( T(x) ) = d( (√[x^2+4])/3 ) - d( x/4 )
6.    d( T(x) ) = (1/3)*d( √[x^2+4] ) - 1/4 dx
7.    d( T(x) ) = (1/3)*( x/√[x^2+4] ) dx - 1/4 dx
8.    d( T(x) )/dx = x/(3√[x^2+4]) - 1/4 = 0

That solves out as:

9.    x = 6/√7 miles

That should minimize the time. Now you need to plug that value back into equation (2) above to figure out the least travel time. 1.941 hours, I think. Do a basic verification by assuming she takes the boat straight to shore, at 2/3 hours, and then walks the 6 miles, at 6/4 hours, and get a total time of 2.167 hours. Then assume she goes entirely by water, for 6.325 miles, dividing that by 3 miles/hr, and also get over 2 hours, too. So perhaps the answer is right. At least it is smaller than either of the two extremes.

But we are talking about only 10-15 minutes difference vs the 2 hour trip, so I think she should do what her mood suggests at the time, unless she is in a hurry.
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