Math Problem Statement

A motorboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:

4

5

6

10

Solution

Let the speed of the motorboat in still water be 1515 km/h. Let the speed of the stream be xx km/h. The total distance downstream and upstream is 30 km each, and the total time taken is 4 hours and 30 minutes, which is equivalent to 4.54.5 hours.

Step 1: Calculate time taken for each part of the trip.

  • Downstream speed = speed of motorboat + speed of stream = 15+x15 + x km/h.
  • Upstream speed = speed of motorboat - speed of stream = 15x15 - x km/h.

The time taken to go downstream is: tdown=3015+xt_{\text{down}} = \frac{30}{15 + x} The time taken to go upstream is: tup=3015xt_{\text{up}} = \frac{30}{15 - x}

Step 2: Set up the equation for total time.

The total time is the sum of the downstream and upstream times: tdown+tup=4.5t_{\text{down}} + t_{\text{up}} = 4.5 Substituting the values: 3015+x+3015x=4.5\frac{30}{15 + x} + \frac{30}{15 - x} = 4.5

Step 3: Solve the equation.

To solve this, first eliminate the fractions by finding a common denominator.

Let's solve it.from sympy import symbols, Eq, solve

Define the variable and equation

x = symbols('x')

Downstream and upstream time equations

downstream_time = 30 / (15 + x) upstream_time = 30 / (15 - x)

Total time equation

total_time_eq = Eq(downstream_time + upstream_time, 4.5)

Solve for x

speed_of_stream = solve(total_time_eq, x) speed_of_streamThe solution for the speed of the stream xx is 55 km/h.

Since speed cannot be negative, the speed of the stream is 5\boxed{5} km/h.

Would you like more details on the steps, or do you have any other questions?

Here are five related questions:

  1. What would the total time be if the stream's speed were 4 km/h?
  2. How would the problem change if the motorboat had a different speed?
  3. What if the distance downstream and upstream were unequal?
  4. How do currents affect real-world navigation for boats?
  5. What are other ways to solve similar riverboat problems?

Tip: When working with distance-rate-time problems, always ensure that the units are consistent across calculations.

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Math Problem Analysis

Mathematical Concepts

Speed, Distance, Time Problems
Relative Speed

Formulas

Time = Distance / Speed

Theorems

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Suitable Grade Level

Grades 9-12