This problem involves concepts from boat and stream relative speed calculations. We are given the speed of the boat in still water, the speed of the water current, and the total time taken for a round trip (to a place and back). We need to find the distance of the place from the starting point.

Let's define the key terms and values:

• Speed of the boat in still water = $v_b = 8$ km/h
• Speed of the water current = $v_c = 2$ km/h
• Total time for the round trip = $T = 2$ hours
• Let the distance to the place be $d$ km.

When the boat travels downstream, the speed of the current adds to the speed of the boat. This is the effective speed when going with the current.

Speed downstream ($v_d$) = Speed of boat in still water + Speed of current

$v_d = v_b + v_c$

$v_d = 8 \text{ km/h} + 2 \text{ km/h} = 10 \text{ km/h}

When the boat travels upstream, the speed of the current opposes the speed of the boat. This is the effective speed when going against the current.

Speed upstream ($v_u$) = Speed of boat in still water - Speed of current

$v_u = v_b - v_c$

$v_u = 8 \text{ km/h} - 2 \text{ km/h} = 6 \text{ km/h}

The problem states that the total time for the round trip (going to the place and coming back) is 2 hours. The time taken to travel a certain distance is given by the formula:

Time = Distance / Speed

Let $t_d$ be the time taken to travel downstream (to the place) and $t_u$ be the time taken to travel upstream (back from the place).

$t_d = \frac{\text{Distance}}{\text{Speed downstream}} = \frac{d}{v_d} = \frac{d}{10}$

$t_u = \frac{\text{Distance}}{\text{Speed upstream}} = \frac{d}{v_u} = \frac{d}{6}$

The total time is the sum of the time taken for the downstream and upstream journeys:

$T = t_d + t_u$

$2 = \frac{d}{10} + \frac{d}{6}$

Now, we need to solve this equation for $d$. To combine the terms on the right side, we find a common denominator for 10 and 6. The least common multiple (LCM) of 10 and 6 is 30.

Multiply both sides of the equation by 30:

$30 \times 2 = 30 \times \left(\frac{d}{10} + \frac{d}{6}\right)$

$60 = 30 \times \frac{d}{10} + 30 \times \frac{d}{6}$

$60 = 3d + 5d$

$60 = 8d$

Now, isolate $d$ by dividing both sides by 8:

$d = \frac{60}{8}$

$d = \frac{30}{4}$

$d = \frac{15}{2}$

$d = 7.5$

So, the distance of the place is 7.5 km.

Understanding Boat and Stream Speed Concepts

In boat and stream problems, we deal with relative speeds. The speed of the boat is affected by the speed of the water current. There are two main scenarios:

• Downstream Motion: When the boat moves in the same direction as the current. The speeds add up. Effective speed = Speed of boat in still water + Speed of current.
• Upstream Motion: When the boat moves against the direction of the current. The current opposes the boat's motion. Effective speed = Speed of boat in still water - Speed of current.

These speed calculations are crucial for determining the time taken to cover a certain distance in either direction.

Step-by-Step Distance Calculation

Let's summarize the steps followed to find the distance:

1. Identify the given information: speed of boat in still water, speed of current, and total round trip time.
2. Calculate the speed downstream by adding the speed of the boat and the speed of the current.
3. Calculate the speed upstream by subtracting the speed of the current from the speed of the boat.
4. Set up an equation using the formula Time = Distance / Speed for both the downstream and upstream journeys. Let the unknown distance be 'd'.
5. The total time is the sum of the time taken for downstream and upstream travel. Form the equation: Total Time = (d / Speed downstream) + (d / Speed upstream).
6. Solve the equation for 'd' to find the distance. This usually involves finding a common denominator and algebraic manipulation.
7. The resulting value of 'd' is the distance of the place.

Let's put the calculated speeds and times in a table:

The equation derived from the total time is: $\frac{d}{10} + \frac{d}{6} = 2$. Solving this equation gives $d = 7.5$ km.

Revision Table: Boat and Stream Formulae

Using these fundamental boat and stream formulae helps solve various problems related to speeds, distances, and times in moving water.

Additional Information on Relative Speed

The concept of relative speed is fundamental to solving boat and stream problems. Relative speed is the speed of an object with respect to another object. In the case of a boat in a current, the boat's speed relative to the water is its speed in still water ($v_b$). However, its speed relative to the ground (what we observe from the shore) depends on the water current's speed ($v_c$).

• When moving downstream, the water's speed adds to the boat's speed, increasing its speed relative to the ground ($v_b + v_c$).
• When moving upstream, the water's speed subtracts from the boat's speed, decreasing its speed relative to the ground ($v_b - v_c$). Note that for upstream travel to be possible, the boat's speed in still water must be greater than the speed of the current ($v_b > v_c$). If $v_b \le v_c$, the boat would be carried downstream by the current even if it's trying to move upstream relative to the water.

Understanding these relative speeds is key to correctly applying the time-distance-speed relationship in boat and stream problems.