Understanding Boat and Stream Concepts

This question involves the concepts of boat speed in still water and the speed of the stream. When a boat moves in a stream, its effective speed changes depending on whether it is moving in the same direction as the stream (downstream) or against the direction of the stream (upstream).

• Speed in Still Water: This is the boat's speed without the influence of any current. Given as 30 km/h.
• Speed of the Stream: This is the speed of the water flow. Given as 7.5 km/h.

Calculating Downstream and Upstream Speeds

When the boat travels downstream, the speed of the stream adds to the boat's speed in still water, making it faster. When it travels upstream, the speed of the stream opposes the boat's motion, making it slower.

Let \(V_b\) be the speed of the boat in still water and \(V_s\) be the speed of the stream.

• Downstream Speed (\(V_d\)): \(V_d = V_b + V_s\)
• Upstream Speed (\(V_u\)): \(V_u = V_b - V_s\)

Given:

• \(V_b = 30 \text{ km/h}\)
• \(V_s = 7.5 \text{ km/h}\)

Calculating the speeds:

• Downstream Speed: \(V_d = 30 \text{ km/h} + 7.5 \text{ km/h} = 37.5 \text{ km/h}\)
• Upstream Speed: \(V_u = 30 \text{ km/h} - 7.5 \text{ km/h} = 22.5 \text{ km/h}\)

Determining the Distance Travelled

The boat travels from point A to B and then back to A, covering a total distance of 90 km. This means the distance from A to B (one way) is half of the total distance.

Total distance = 90 km (A to B and B to A)

Distance from A to B = Distance from B to A = \(\frac{90 \text{ km}}{2} = 45 \text{ km}\)

Calculating Time Taken for Each Leg of the Journey

The formula for time is:

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

We need to calculate the time taken for the downstream journey (say, A to B) and the time taken for the upstream journey (B to A).

• Time Downstream (\(T_d\)): The distance is 45 km and the speed is 37.5 km/h.
• \(T_d = \frac{45 \text{ km}}{37.5 \text{ km/h}}\)

Let's calculate \(T_d\):

\(T_d = \frac{45}{37.5} = \frac{450}{375}\)

Dividing both numerator and denominator by 75:

\(450 \div 75 = 6\)

\(375 \div 75 = 5\)

So, \(T_d = \frac{6}{5} = 1.2 \text{ hours}\)

• Time Upstream (\(T_u\)): The distance is 45 km and the speed is 22.5 km/h.
• \(T_u = \frac{45 \text{ km}}{22.5 \text{ km/h}}\)

Let's calculate \(T_u\):

\(T_u = \frac{45}{22.5} = \frac{450}{225}\)

Dividing both numerator and denominator by 225:

\(450 \div 225 = 2\)

\(225 \div 225 = 1\)

So, \(T_u = \frac{2}{1} = 2 \text{ hours}\)

Calculating Total Time for the Round Trip

The total time taken for the round trip from A to B and back to A is the sum of the time taken for the downstream journey and the time taken for the upstream journey.

\(\text{Total Time} (T_{\text{total}}) = T_d + T_u\)

\(T_{\text{total}} = 1.2 \text{ hours} + 2 \text{ hours}\)

\(T_{\text{total}} = 3.2 \text{ hours}\)

The total time taken by the boat to cover a total distance of 90 km (A to B and back to A) is 3.2 hours.

The final answer is 3.2 hours.

Revision Table: Boat and Stream Basics

Additional Information: Distance, Speed, and Time Problems

Problems involving boat and stream speed are a specific type of distance, speed, and time problem. The key is to correctly identify the effective speed of the object (boat, swimmer, etc.) relative to the ground when influenced by a moving medium (stream, wind, etc.).

• Always determine the speeds relative to the medium (still water, still air) and the speed of the medium itself.
• Combine these speeds appropriately for 'with' the medium (downstream/tailwind) and 'against' the medium (upstream/headwind) scenarios.
• Use the standard distance, speed, time formula (\(D=S \times T\) or \(T = \frac{D}{S}\) or \(S = \frac{D}{T}\)).
• Pay attention to whether the distance given is for a one-way trip or a round trip.