A man rows a boat at a speed of 15 mph in still water. Find the speed of the river if it takes her 4 hours 30 minutes to row a boat to a place 30 miles away and return.

5 mph

10 mph

12 mph

20 mph

To solve this problem, we need to account for the effect of the river's current on the rowing speed. Let's denote:

- \( v_b \) = speed of the boat in still water = 15 mph
- \( v_r \) = speed of the river (current)
- \( d \) = distance to the destination = 30 miles
- \( t \) = total time for the round trip = 4 hours 30 minutes = 4.5 hours

**Downstream Speed**: When rowing downstream, the speed of the boat relative to the ground is \( v_b + v_r \).

**Upstream Speed**: When rowing upstream, the speed of the boat relative to the ground is \( v_b - v_r \).

### Calculating Time for Each Leg of the Trip
The total time taken for the trip is the sum of the time taken downstream and upstream:

\[ t = \frac{d}{v_b + v_r} + \frac{d}{v_b - v_r} \]

Plugging in the known values:

\[ 4.5 = \frac{30}{15 + v_r} + \frac{30}{15 - v_r} \]

To find \( v_r \), solve this equation:

\[ 4.5 = \frac{30}{15 + v_r} + \frac{30}{15 - v_r} \]

\[ 4.5 = \frac{30(15 - v_r) + 30(15 + v_r)}{(15 + v_r)(15 - v_r)} \]

\[ 4.5 = \frac{450 - 30v_r + 450 + 30v_r}{225 - v_r^2} \]

\[ 4.5 = \frac{900}{225 - v_r^2} \]

\[ 225 - v_r^2 = \frac{900}{4.5} \]

\[ 225 - v_r^2 = 200 \]

\[ v_r^2 = 225 - 200 \]

\[ v_r^2 = 25 \]

\[ v_r = 5 \]

The speed of the river is 5 mph.

The answer is: **5 mph**.

Would you like more details or have any questions? Here are some follow-up questions you might consider:

1. How would the time change if the boat speed increased?
2. How does the speed of the river affect the total time taken?
3. What is the formula for calculating the effective speed in moving water?
4. How do we calculate the speed of the boat relative to the river's current?
5. What happens if the river's speed exceeds the boat's speed?

**Tip:** When solving problems involving moving water and a boat, always consider the relative speeds and the impact of the current in both directions.

Learn how to calculate the speed of a river current based on a boat's speed and travel time. This problem involves understanding the relationship between boat speed, river current, and total travel time. Suitable for Grades 7-9.

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