Two ducks move along the circumference of a circular pond in the same direction and come alongside each other every 54 minutes. If they moved with the same speeds in the opposite directions, they would meet every 9 minutes. It is known that when the ducks moved along the circumference in opposite directions, the distance between them decreased from 54 to 14 feet every 48 seconds. What is the speed of the slower duck?

Options

A. A)20 feet/min
B. B)15 feet/min
C. C)30 feet/min
D. D)20.83 feet/min

Answer

To solve the problem step by step, we will use the information given about the ducks and their movements.

### Step 1: Understand the Problem
We have two ducks moving around a circular pond. When they move in the same direction, they meet every 54 minutes. When they move in opposite directions, they meet every 9 minutes.

### Step 2: Define Variables
Let:
- \( S_1 \) = speed of the faster duck (in feet per minute)
- \( S_2 \) = speed of the slower duck (in feet per minute)

### Step 3: Calculate the Combined Speed in Opposite Directions
When the ducks move in opposite directions, their speeds add up. They meet every 9 minutes, so the distance covered in that time is equal to the sum of their speeds multiplied by the time:
\[ S_1 + S_2 = \text{Distance covered in 9 minutes} \]
The distance between them decreases from 54 feet to 14 feet in 48 seconds. The distance covered in that time is:
\[ 54 - 14 = 40 \text{ feet} \]
Thus, the combined speed of the ducks is:
\[ \text{Speed} = \frac{40 \text{ feet}}{48 \text{ seconds}} = \frac{40 \text{ feet}}{0.8 \text{ minutes}} = 50 \text{ feet per minute} \]
So, we have:
\[ S_1 + S_2 = 50 \text{ feet per minute} \quad (1) \]

### Step 4: Calculate the Relative Speed in Same Direction
When the ducks move in the same direction, they meet every 54 minutes. The distance they cover in that time is equal to the difference of their speeds multiplied by the time:
\[ S_1 - S_2 = \frac{50 \text{ feet}}{54 \text{ minutes}} \times 54 = 25 \text{ feet per minute} \quad (2) \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( S_1 + S_2 = 50 \)
2. \( S_1 - S_2 = 25 \)

We can add these two equations to eliminate \( S_2 \):
\[ (S_1 + S_2) + (S_1 - S_2) = 50 + 25 \]
This simplifies to:
\[ 2S_1 = 75 \implies S_1 = 37.5 \text{ feet per minute} \]

Now, substitute \( S_1 \) back into equation (1):
\[ 37.5 + S_2 = 50 \implies S_2 = 50 - 37.5 = 12.5 \text{ feet per minute} \]

### Step 6: Conclusion
The speed of the slower duck \( S_2 \) is 12.5 feet per minute.