A man can row at 30 km/h, and run at 16 km/h. He needs to get from a point A, on the south bank of a stretch of still water, to point B on the north bank of the water. The direct distance from A to B is 34 km, and the water is 30 km wide. He starts rowing with an angle "theta" between North and the direction in which he rows. Find an expression for the time T he will take to get from A to B, in terms of "theta".
T(theta) =
T(theta) =
-
Let point C be the point on the north bank that is directly due north of point A.
By Pythagoras
CB = root (34^2 - 30^2) = 16 km
The track across the water will be 30 / Cos(theta) and the time taken to cross the water will be
Tw = (30/Cos(theta))/30 = 1/Cos(theta) hours
Once he reaches the north bank, he will need to run
abs(16 - 30Tan(theta)) along the bank to reach B this will take him
Tb = abs(16 - 30Tan(theta))/16 = abs(1 - 1.875Tan(theta)) hours
The total time T to go from A to B is
T = Tw + Tb
T(theta) = 1/Cos(theta) + abs(1 - 1.875Tan(theta)) hours.
I have used abs to signify the value of the expression without its sign. e.g. abs(-2) would be 2. This is because the landing point after crossing the water could be to the east or west of point B, and he will have to run from there towards B.
The expression above assumes that theta is positive if the direction in which he rows (east or west) is the same as the direction of point B. If the direction he rows is the other way, then theta will negative. The expression will still work.
The value of theta must be -90 < theta < 90
By Pythagoras
CB = root (34^2 - 30^2) = 16 km
The track across the water will be 30 / Cos(theta) and the time taken to cross the water will be
Tw = (30/Cos(theta))/30 = 1/Cos(theta) hours
Once he reaches the north bank, he will need to run
abs(16 - 30Tan(theta)) along the bank to reach B this will take him
Tb = abs(16 - 30Tan(theta))/16 = abs(1 - 1.875Tan(theta)) hours
The total time T to go from A to B is
T = Tw + Tb
T(theta) = 1/Cos(theta) + abs(1 - 1.875Tan(theta)) hours.
I have used abs to signify the value of the expression without its sign. e.g. abs(-2) would be 2. This is because the landing point after crossing the water could be to the east or west of point B, and he will have to run from there towards B.
The expression above assumes that theta is positive if the direction in which he rows (east or west) is the same as the direction of point B. If the direction he rows is the other way, then theta will negative. The expression will still work.
The value of theta must be -90 < theta < 90