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:

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4

5

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10

Let the speed of the motorboat in still water be \( 15 \) km/h. Let the speed of the stream be \( x \) 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 \( 4.5 \) hours.

### Step 1: Calculate time taken for each part of the trip.
- **Downstream speed** = speed of motorboat + speed of stream = \( 15 + x \) km/h.
- **Upstream speed** = speed of motorboat - speed of stream = \( 15 - x \) km/h.

The time taken to go downstream is:
\[
t_{\text{down}} = \frac{30}{15 + x}
\]
The time taken to go upstream is:
\[
t_{\text{up}} = \frac{30}{15 - x}
\]

### Step 2: Set up the equation for total time.
The total time is the sum of the downstream and upstream times:
\[
t_{\text{down}} + t_{\text{up}} = 4.5
\]
Substituting the values:
\[
\frac{30}{15 + x} + \frac{30}{15 - x} = 4.5
\]

### 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 \( x \) is \( 5 \) km/h. 

Since speed cannot be negative, the speed of the stream is \( \boxed{5} \) 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.

Learn how to calculate the speed of the stream in a riverboat problem where a motorboat travels downstream and upstream at different speeds. Detailed step-by-step solution provided using the concept of relative speed and time calculations. Suitable for students in Grades 9-12.

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