A person in a rowboat 2 miles from the nearest point P on a straight shoreline wishes to reach a house 6 miles down the shoreline from P.
If the person can row at a rate of 3 miles per hour and walk at a rate of 5 miles per hour, find the least amount of time required to reach the house.
The nearest distance of the person to the spot P on the shore = 2miles.
Let starting point of the person to row the boat be S. Then SP = 2.
Let the house be at the spot H on the bank of the river.
Then SP = 2, SH = 6 miles.
Therefore PSH form a right angled triangle with right angle at S.
Therefore PS^2+SH^2 = PH^2 by Pyhagoras theorem.
2^2+6^2 = PH^2.
Or PH^2 = 2^2+6^2 = 4+36 = 40
Therefore PH = sqrt(40) = 6.3246 miles.
Therefore the time required to row SH = 6 miles at the rate of 3mph = 6/3 = 2 hours.
The time required to walk the distance of PH = 6.3246 miles at the rate of 5 miles per hour = 6.3246/5 = 1.2649 hours = 1 hour 15.9 minutes nealy.
So the least time to reach is the option of walking.