A tank of `4800m^(3)` capacity is full of water. The discharging capacity of the pump is `10m^(3)`/min higher than its filling capacity. As a result the pump needs 16 min less to discharge the fuel than to fill up the tank. Find the filling capacity of the pump.

Options

A. `50m^(3)//`min
B. `25m^(3)//`min
C. `55m^(3)//`min
D. `24m^(3)//`min

Answer

To solve the problem step by step, we will define variables and set up equations based on the information given in the question.

### Step 1: Define Variables
Let the filling capacity of the pump be \( x \) m³/min.

### Step 2: Define Discharging Capacity
According to the question, the discharging capacity of the pump is 10 m³/min higher than the filling capacity. Therefore, the discharging capacity will be:
\[ x + 10 \text{ m³/min} \]

### Step 3: Calculate Time to Fill the Tank
The time taken to fill the tank can be calculated using the formula:
\[ \text{Time to fill} = \frac{\text{Capacity of tank}}{\text{Filling capacity}} = \frac{4800}{x} \text{ minutes} \]

### Step 4: Calculate Time to Discharge the Tank
Similarly, the time taken to discharge the tank is:
\[ \text{Time to discharge} = \frac{\text{Capacity of tank}}{\text{Discharging capacity}} = \frac{4800}{x + 10} \text{ minutes} \]

### Step 5: Set Up the Equation
According to the problem, the time taken to discharge the tank is 16 minutes less than the time taken to fill the tank. Therefore, we can set up the equation:
\[ \frac{4800}{x} - \frac{4800}{x + 10} = 16 \]

### Step 6: Simplify the Equation
To simplify the equation, we will find a common denominator:
\[ \frac{4800(x + 10) - 4800x}{x(x + 10)} = 16 \]
This simplifies to:
\[ \frac{48000}{x(x + 10)} = 16 \]

### Step 7: Cross Multiply
Cross multiplying gives us:
\[ 48000 = 16x(x + 10) \]
This simplifies to:
\[ 48000 = 16x^2 + 160x \]

### Step 8: Rearrange the Equation
Rearranging the equation gives us:
\[ 16x^2 + 160x - 48000 = 0 \]
Dividing the entire equation by 16 simplifies it to:
\[ x^2 + 10x - 3000 = 0 \]

### Step 9: Factor the Quadratic Equation
Now we need to factor the quadratic equation:
\[ x^2 + 60x - 50x - 3000 = 0 \]
Factoring gives:
\[ (x - 50)(x + 60) = 0 \]

### Step 10: Solve for x
Setting each factor to zero gives us two potential solutions:
1. \( x - 50 = 0 \) → \( x = 50 \)
2. \( x + 60 = 0 \) → \( x = -60 \)

Since the filling capacity cannot be negative, we discard \( x = -60 \).

### Final Answer
Thus, the filling capacity of the pump is:
\[ x = 50 \text{ m³/min} \]