Solving Boat and Stream Problems: Finding Downstream Speed

This problem involves a boat traveling both upstream (against the current) and downstream (with the current). The speed of the boat relative to the water changes depending on whether it's going with or against the stream. We are given distances, total time, and the speed of the stream, and we need to find the boat's speed when going downstream.

Defining Key Terms and Variables

• Speed of the boat in still water: Let this be \( u \) km/h. This is the boat's speed without any influence from the stream.
• Speed of the stream: Given as \( v = 5 \) km/h.
• Speed upstream: When the boat goes against the stream, the stream's speed reduces the boat's effective speed. Speed upstream \( = u - v = u - 5 \) km/h.
• Speed downstream: When the boat goes with the stream, the stream's speed adds to the boat's effective speed. Speed downstream \( = u + v = u + 5 \) km/h.

Setting up the Time-Distance-Speed Relationship

The fundamental relationship between time, distance, and speed is:

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

We are given the total time taken for the entire journey (upstream and downstream).

• Distance upstream \( = 10 \) km
• Distance downstream \( = 11 \) km
• Total time \( = 52 \) minutes

First, let's convert the total time from minutes to hours, as the speeds are given in km/h:

\( 52 \text{ minutes} = \frac{52}{60} \text{ hours} = \frac{13}{15} \text{ hours} \)

Now, we can express the time taken for each part of the journey:

• Time upstream \( = \frac{\text{Distance upstream}}{\text{Speed upstream}} = \frac{10}{u - 5} \) hours
• Time downstream \( = \frac{\text{Distance downstream}}{\text{Speed downstream}} = \frac{11}{u + 5} \) hours

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

\( \frac{10}{u - 5} + \frac{11}{u + 5} = \frac{13}{15} \)

Solving the Equation to Find the Speed of the Boat in Still Water

We now have an equation involving \( u \). Let's solve for \( u \):

Combine the terms on the left side by finding a common denominator, which is \( (u - 5)(u + 5) = u^2 - 25 \):

\( \frac{10(u + 5) + 11(u - 5)}{(u - 5)(u + 5)} = \frac{13}{15} \)

\( \frac{10u + 50 + 11u - 55}{u^2 - 25} = \frac{13}{15} \)

\( \frac{21u - 5}{u^2 - 25} = \frac{13}{15} \)

Now, cross-multiply:

\( 15(21u - 5) = 13(u^2 - 25) \)

\( 315u - 75 = 13u^2 - 325 \)

Rearrange the terms to form a quadratic equation:

\( 13u^2 - 315u - 325 + 75 = 0 \)

\( 13u^2 - 315u - 250 = 0 \)

This is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 13 \), \( b = -315 \), and \( c = -250 \). We can use the quadratic formula to solve for \( u \):

\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Calculate the discriminant \( \Delta = b^2 - 4ac \):

\( \Delta = (-315)^2 - 4(13)(-250) \)

\( \Delta = 99225 - (-13000) \)

\( \Delta = 99225 + 13000 \)

\( \Delta = 112225 \)

Now, find the square root of the discriminant:

\( \sqrt{\Delta} = \sqrt{112225} = 335 \)

Now, substitute the values into the quadratic formula:

\( u = \frac{-(-315) \pm 335}{2(13)} \)

\( u = \frac{315 \pm 335}{26} \)

We get two possible values for \( u \):

• \( u_1 = \frac{315 + 335}{26} = \frac{650}{26} = 25 \)
• \( u_2 = \frac{315 - 335}{26} = \frac{-20}{26} \)

Since speed cannot be negative, we discard the second solution. The speed of the boat in still water is \( u = 25 \) km/h.

Calculating the Speed Downstream

The question asks for the speed of the boat when going downstream. We defined the speed downstream as \( u + v \).

Speed downstream \( = u + v = 25 + 5 = 30 \) km/h.

The speed of the boat when going downstream is 30 km/h.

The calculated total time matches the given total time, confirming our value for \( u \) is correct.

Revision Table: Boat and Stream Concepts

Additional Information: Variations in Boat and Stream Problems

Boat and stream problems often involve different types of questions. Understanding the basic formulas is key to solving them. Common variations include:

• Finding the speed of the boat in still water when upstream and downstream speeds are given.
• Finding the speed of the stream when boat speed in still water and upstream/downstream speeds are given.
• Calculating distances covered upstream or downstream given speeds and time.
• Problems involving round trips (going upstream and returning downstream) with a total time given.
• Problems where the time taken for upstream and downstream journeys is related (e.g., time upstream is twice the time downstream).

These problems typically rely on setting up equations based on the time = distance / speed relationship and solving for the unknown variable, often leading to linear or quadratic equations.