The speed of a boat in still water is 8 km/hr It can go 15 km upstream and 22 km downstream in 5 hours. The speed of the stream is ----- km/hr

The speed of a boat in still water is 8 km/hr It can go 15 km upstream and 22 km downstream in 5 hours. The speed of the stream is ----- km/hr

When dealing with problems related to the speed of a boat in a stream, we often need to consider the effects of the current. This involves calculating the boat's speed both upstream and downstream.

Speed of the boat in still water = 8 km/hr

Distance upstream = 15 km

Distance downstream = 22 km

Total time = 5 hours

Let x be the speed of the stream

Upstream speed = 8−x km/hr

Downstream speed = 8+x km/hr

Time to travel upstream = 15/8−x hours

Time to travel downstream = 22/8+x hours

Total time = Time upstream + Time downstream

15/(8−x) + 22/(8+x) = 5

Combine the fractions over a common denominator and solve for x

120+15x+176−22x = 5(64−x²),

296−7x = 320−5x²,

5x²−7x−24=0

Factor the quadratic equation

(x−3)(5x+8) = 0

x = 3 or x = -8/5

Since speed cannot be negative, the speed of the stream is 3 km/hr.

Basics of Boat and Stream Problems

Boat and stream problems in mathematics involve calculating the speed and travel times of boats navigating in rivers or streams. These problems are common in exams and help in understanding the concepts of relative speed, which is a crucial aspect in physics and motion studies.

Speed in Still Water:

• This is the speed of the boat when there is no current. It is the inherent speed of the boat.

Speed of the Stream:

• This is the speed at which the water in the stream or river flows. It can affect the boat's speed positively or negatively.

Upstream and Downstream:

• Upstream: When a boat travels against the direction of the current. The effective speed decreases.
• Downstream: When a boat travels with the direction of the current. The effective speed increases.

Formulas

• Speed Upstream = Speed in Still Water - Speed of the Stream.
• Speed Downstream = Speed in Still Water + Speed of the Stream.
• Time = Distance / Speed.

Steps to Solve Boat and Stream Problems

• Determine the speed of the boat in still water, the speed of the stream, distances to be covered, and the total time if given.
• Use the formulas for upstream and downstream speeds to set up equations based on the problem's requirements.
• Solve the equations to find the unknown values, which could be the speed of the stream, the speed of the boat in still water, or the travel times.
• Ensure that the solutions are realistic. For example, speeds cannot be negative.