Calculus/Maximum-minimum problem - rowing & walking
Tony B. wrote at 2010-11-29 03:28:09
A simple solution with no calculus necessary...
The time it would take to row across the diameter of the lake uses D=RT, or 4 miles = (2mi/hr)(T) per the given. Clearly T = 2 hrs to row across the lake.
The time it would take to walk alongside the lake's perimeter is the same as walking along a semicircle with radius of 2 miles. So, again using D=RT, D=C/2 =(pi)r =(3.14)2 = 6.28 miles along the semicircle to reach the other side. Therefore, the time it takes to walk along the semicircle of 6.28 miles is found by using 6.28 miles = (4mi/hr)(T) per the given. So, T =1.57 hrs to walk alongside the lake's perimeter- less time than rowing a shorter distance because you're moving twice as fast by foot.
Conclusion: Since it takes longer to row across-(2 hrs) than walking along the perimeter- (1.57 hrs), to break it up by rowing part of the way and then walking the remainder of the way would only yield a total time greater than 1.57 hrs, or walking only. Therefore to minimize travel time, one should walk all the way to the other side. Though the calculus driven solutions suggested earlier may seem like overkill, there is still something majestic in seeing the calculus reconcile to simple laws of arithmetic and algebra 1.
Professor A wrote at 2014-12-26 20:59:23
The smallest ratio of a segment of a circle to its intercepted arc occurs when you take the ratio of the diameter of the circle to its intercepted arc which is a half circle. This ratio is 2r to (pi)(r)or 0.6366... This means that if the ratio of the swimming speed to the walking speed is less than 0.6366...the only solution is to walk the entire distance. In this case, the ratio is 0.5 so its a no-brainer. Walk around to the other side.
You will find that if the walking speed is 4 and the swimming speed is 4(0.6366...)= 2.546479..., the time swimming across would be exactly the same as the time walking around.