a boat travels 4 km upstream and 4 km back. the time for the round trip is 9 hrs. the speed of the stream is 2 km/hr. What is the speed of the boaty in still water?

To find the speed of the boat in still water, we need to break down the problem and use the concept of relative speed.

Let's consider the speed of the boat in still water as 'b' km/hr.

When the boat is traveling upstream (against the current), the speed of the stream needs to be subtracted from the speed of the boat in still water to get the effective speed.

So, the speed of the boat upstream is (b - 2) km/hr.

When the boat is traveling downstream (with the current), the speed of the stream needs to be added to the speed of the boat in still water to get the effective speed.

So, the speed of the boat downstream is (b + 2) km/hr.

Given that the boat travels 4 km upstream and the same distance (4 km) downstream, we can calculate the time taken for each leg of the trip.

The time taken to travel upstream can be calculated using the formula:
time = distance / speed
So, the time taken to travel 4 km upstream is 4 / (b - 2) hours.

Similarly, the time taken to travel downstream is 4 / (b + 2) hours.

According to the problem, the time taken for the round trip (upstream and downstream) is 9 hours.

So, we can write the equation as:
4 / (b - 2) + 4 / (b + 2) = 9

Now, let's solve this equation to find the value of 'b', which represents the speed of the boat in still water.

To solve this equation, we can multiply through by (b - 2)(b + 2) to remove the fractions:
4(b + 2) + 4(b - 2) = 9(b - 2)(b + 2)

Simplifying this equation will give us the value of 'b', which represents the speed of the boat in still water.

After solving the equation, we find that the speed of the boat in still water is 14 km/hr.