a motorboat takes 5 hours to travel 45km downstream and then 18 km upstream on a river whose current flow of the rate of 3 km/hr. how fast can the boat travel in still water?
To find the speed of the boat in still water, we need to analyze the given information. Let's break down the problem step by step.
1. Let's consider the speed of the boat in still water as 'b' km/hr, and the rate of the river's current as 'c' km/hr.
2. When the boat is moving downstream (in the same direction as the river's current), its effective speed is the sum of its speed in still water and the speed of the current.
Downstream speed = boat speed in still water + rate of the current
Downstream speed = b + c km/hr
3. In this case, the boat takes 5 hours to travel 45 km downstream. So we can set up the equation:
Distance = Speed × Time
45 km = (b + c) km/hr × 5 hr
4. Next, let's consider the scenario when the boat is moving upstream (opposite to the direction of the river's current). In this case, the boat's effective speed is the difference between its speed in still water and the speed of the current.
Upstream speed = boat speed in still water - rate of the current
Upstream speed = b - c km/hr
5. The boat takes 5 hours to travel 18 km upstream, so we can set up a similar equation:
18 km = (b - c) km/hr × 5 hr
Now we have a system of equations to solve. Let's solve them simultaneously to find the value of 'b' which represents the boat's speed in still water.
From the equation in step 3, we have:
45 = (b + c) × 5
Dividing both sides by 5, we get:
9 = b + c -- Equation 1
From the equation in step 5, we have:
18 = (b - c) × 5
Dividing both sides by 5, we get:
3.6 = b - c -- Equation 2
Now, we can solve this system of equations by adding or subtracting them:
Adding Equation 1 and Equation 2, we get:
9 + 3.6 = b + c + b - c
Simplifying the equation:
12.6 = 2b
Dividing both sides by 2, we get:
b = 6.3
Therefore, the boat can travel at a speed of 6.3 km/hr in still water.