Can anyone help me solve this Math’s question from Vectors please? 10 points for correct answer?
3. A river flows at 5 m/s from the west to east between parallel banks which are a distance of 300 m apart. A man rows a boat at a speed of 3 m/s in still water.
a) State the direction in which the boat must be steered in order to cross the river from the southern bank to the northern bank in the shortest possible time.
b) Find the time taken and the actual distance covered by the boat in this crossing.
- MathmomLv 71 decade agoFavorite Answer
Since the river flows from west to east, its angle is 0°
Let Va be vector river flows in 1 sec. Then
Va = (5*cos0°, 5*sin0°) = (5, 0)
Let θ be the angle man rows boat across river (assume he is crossing from south to north).
Let Vb be vector of man rowing boat across river. Then
Vb = (3*cosθ, 3*sinθ)
Let Vc be vector of boat crossing river. Then
Vc = Va + Vb = (5, 0) + (3*cosθ, 3*sinθ) = (5+3*cosθ, 3*sinθ)
Now since we want shortest possible time to cross river, we need to maximize vertical distance (y-value of Vc) traveled each second, i.e 3*sinθ.
This is maximized when sinθ is maximized, i.e. sinθ = 1 or θ = 90°.
So we can calculate
Vb = (3*cos90°, 3*sin90°) = (0,3)
so each second boat travels
Vc = Va + Vb = (5, 0) + (0,3) = (5,3) (5 meters E, 3 meters N)
Since we want to travel 300 meters N, it will take 300/3 = 100 sec.
In 100 seconds, boat has traveled 500 meters E, 300 meters N, so total distance covered by boat
= sqrt (500^2 + 300^2) = sqrt (340000) ≈ 583.1 meters