Question

A motorboat covers a certain distance downstream, in a river in 3 hours. It covers the same distance upstream in 3 hours and a half. If the speed of water is 1.5 km/hr, then the speed of the boat in still water is:(a) 17 km/hr(b) 19.5 km/hr(c) 17.5 km/hr(d) 19 km/hr

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Hint: First of all, we will assume some variable for the speed of the boat and the distance it covers. We shall keep in mind that when moving downstream, the speed of the river adds to the speed of the boat and when moving upstream, the water is reduced from the speed of the boat. We shall use the speed formula from the physics to form equations which can be solved for the values of speed and distance. The formula of speed is s = $\dfrac{l}{t}$, where s is the speed, l is the distance it covers and t is the time required to cover the distance of l at speed s.

Complete step-by-step solution:
Let the speed of the boat be x and the distance it covers be l.
It is given to us that when moving downstream, it takes 3 hours.
Therefore, t = 3 hours. The net speed downstream will be the sum of the speed of the boat and the speed of the river.
Thus, net speed downstream = s = x + 1.5
So, we will use equation of speed.
\begin{align} & \Rightarrow x+1.5=\dfrac{l}{3} \\ & \Rightarrow 3x-l=-4.5......\left( 1 \right) \\ \end{align}
Now, when the boat is going upstream, it takes 3.5 hours.
Therefore, t = 3.5 hours. The net speed upstream will be the sum of the speed of the boat and the speed of the river.
Thus, net speed upstream = s = x – 1.5
So, we will use equation of speed.
\begin{align} & \Rightarrow x-1.5=\dfrac{l}{3.5} \\ & \Rightarrow 3.5x-l=5.25......\left( 2 \right) \\ \end{align}
Now, we shall solve (1) and (2).
Subtract (2) from (1).
\begin{align} & \Rightarrow 0.5x=5.25+4.5 \\ & \Rightarrow 0.5x=9.75 \\ & \Rightarrow x=19.5 \\ \end{align}
Thus, the speed of the boat = x = 19.5 km/hr.
Hence, option (b) is the correct option.

Note: This problem is a classic example of an equation in two variables. We’ve solved the equations with the elimination method. It can also be solved by other methods like completing the square method or cross multiplication method. Moreover, we are not asked to find the length. Hence, we concluded the solution as soon as we found speed.