Math Problem Statement

The speed of a boat in still water is 9 km/h. Length of the river is 36 km. The boat takes 9 hours to go and come back. Find the speed of the stream.

Solution

Let's solve the problem step-by-step.

Given:

  1. Speed of the boat in still water: vb=9v_b = 9 km/hr.
  2. Distance of the river (one way): D=36D = 36 km.
  3. Total time taken for the journey (going and coming back): T=9T = 9 hours.

To Find:

  • Speed of the stream vsv_s.

Steps to Solve:

  1. Define the Speeds in Each Direction:

    • Downstream speed (going with the stream):
      The speed of the boat downstream is the sum of the boat's speed in still water and the stream's speed: vdown=vb+vs=9+vskm/hrv_{\text{down}} = v_b + v_s = 9 + v_s \, \text{km/hr}

    • Upstream speed (going against the stream):
      The speed of the boat upstream is the difference between the boat's speed in still water and the stream's speed: vup=vbvs=9vskm/hrv_{\text{up}} = v_b - v_s = 9 - v_s \, \text{km/hr}

  2. Calculate Time for Each Leg of the Journey:

    • Time taken downstream (from start to end of river):
      tdown=Dvdown=369+vst_{\text{down}} = \frac{D}{v_{\text{down}}} = \frac{36}{9 + v_s}

    • Time taken upstream (from end back to start):
      tup=Dvup=369vst_{\text{up}} = \frac{D}{v_{\text{up}}} = \frac{36}{9 - v_s}

  3. Set Up the Total Time Equation:

    The total time for the journey (going downstream and coming back upstream) is the sum of both times: tdown+tup=Tt_{\text{down}} + t_{\text{up}} = T Substituting the expressions for tdownt_{\text{down}} and tupt_{\text{up}}: 369+vs+369vs=9\frac{36}{9 + v_s} + \frac{36}{9 - v_s} = 9

  4. Solve for vsv_s:

    First, divide the entire equation by 9 to simplify: 49+vs+49vs=1\frac{4}{9 + v_s} + \frac{4}{9 - v_s} = 1

    Clear the denominators by multiplying through by (9+vs)(9vs)(9 + v_s)(9 - v_s): 4(9vs)+4(9+vs)=(9+vs)(9vs)4(9 - v_s) + 4(9 + v_s) = (9 + v_s)(9 - v_s)

    Expand both sides: 364vs+36+4vs=81vs236 - 4v_s + 36 + 4v_s = 81 - v_s^2

    Combine like terms: 72=81vs272 = 81 - v_s^2

    Rearrange to isolate vs2v_s^2: vs2=8172v_s^2 = 81 - 72 vs2=9v_s^2 = 9

    Take the square root of both sides: vs=9=3v_s = \sqrt{9} = 3

Answer:

The speed of the stream is 3 km/hr.

Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:

  1. What would be the total time taken if the speed of the stream increased to 4 km/hr?
  2. How would the total journey time change if the river length was doubled to 72 km?
  3. If the boat took 10 hours instead of 9 hours for the same journey, what would the stream's speed be?
  4. Can you determine the speed of the boat in still water if the stream's speed was 5 km/hr and the total time was 8 hours?
  5. What happens to the upstream time if the stream's speed equals the boat's speed in still water?

Tip: When solving problems involving relative motion in water, always distinguish between upstream and downstream conditions to avoid confusion.

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Math Problem Analysis

Mathematical Concepts

Speed, Distance, and Time Calculations
Relative Motion in Rivers

Formulas

Time = Distance / Speed
Speed of Boat in Still Water = Downstream Speed - Stream Speed
Speed of Boat in Still Water = Upstream Speed + Stream Speed

Theorems

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Suitable Grade Level

Grades 9-12