Shortest Time
A man is on an island which is 4 miles from the nearest point on a straight shoreline. He wishes to go to a house which is 12 miles from this nearest point. If he rows at 3 miles per hour and runs at 5 miles per hour, find the shortest time to reach the house.If the quickest time is achieved when x =3, how much time does he waste if he lands at x =2 or x =4?
Labels: mathemagic, SharedPuzzle
posted by Ragknot at
3:21 PM
4 Comments:
the ~12.65 miles way... its the quickest
yah dats rite im da firs one... XD
The path of the man can be described with two lengths, the distance traveled on the water and the distance traveled on land.
r(x)=sqrt(x²+16)
l(x)=12-x
To find the time it took to travel the distance, we divide by the speed at which he goes at.
sqrt(x²+16)/3
(12-x)/5
if we add the two formulas, we get the time took for the entire trip.
f(x)=sqrt(x²+16)/3 + (12-x)/5
We want to minimize f(x) so to find the absolute minimum we take the derivative
f(x)= sqrt(x²+16)/3 + (12-x)/5
f'(x)= 1/3[1/2(x²+16)^(-1/2)*(2x)] - 1/5
f'(x)= x/(3sqrt(x²+16)) - 1/5
the slope is 0 when at the minimum
0 = x/(3sqrt(x²+16)) - 1/5
1/5 = x/(3sqrt(x²+16))
1/(5x) = 1/(3sqrt(x²+16))
5x = 3sqrt(x²+16)
25x² = 9(x²+16)
25x² = 9x² + 144
16x² = 144
x = 3
Therefore when x=3 the time taken will be at its minimum. We substitute it back in to find the time.
f(x)=sqrt(x²+16)/3 + (12-x)/5
f(3)=sqrt(3²+16)/3 + (12-3)/5
f(3)=5/3 + 9/5
f(3)=52/15
The shortest time needed to travel is 3.47 hours or 3 hours and 28 minutes.
Follow up Question
If the quickest time is when X = 3 miles, how much time has he wasted if he lands at X=2 miles or X=4miles?
To Sam,
very good answer.
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