Two tanks A and B contain water at `30^@C` and `80^@C` respectively calculate the amount of water that must be taken from each tank respectively to prepare 40 kg of water at `50^@C`

Options

A. 24 kg, 16 kg
B. 16 kg, 24kg
C. 20 kg, 20kg
D. 30 kg, 10 kg

Answer

To solve the problem of mixing water from two tanks A and B to achieve a desired temperature, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = amount of water taken from tank A (at \( 30^\circ C \))
- \( y \) = amount of water taken from tank B (at \( 80^\circ C \))

### Step 2: Set Up the Equation for Total Mass
We know that the total mass of the mixture should be \( 40 \, kg \):
\[ x + y = 40 \quad \text{(Equation 1)} \]

### Step 3: Apply the Principle of Calorimetry
According to the principle of calorimetry, the heat gained by the cooler water equals the heat lost by the warmer water. The heat gained by water from tank A can be expressed as:
\[ \text{Heat gained by A} = x \cdot c \cdot (50 - 30) = x \cdot c \cdot 20 \]
The heat lost by water from tank B can be expressed as:
\[ \text{Heat lost by B} = y \cdot c \cdot (80 - 50) = y \cdot c \cdot 30 \]
Here, \( c \) is the specific heat capacity of water, which is the same for both tanks and will cancel out.

### Step 4: Set Up the Heat Gain and Loss Equation
Setting the heat gained equal to the heat lost gives us:
\[ x \cdot 20 = y \cdot 30 \quad \text{(Equation 2)} \]

### Step 5: Solve Equation 2 for One Variable
From Equation 2, we can express \( x \) in terms of \( y \):
\[ x = \frac{30}{20}y = \frac{3}{2}y \]

### Step 6: Substitute into Equation 1
Substituting \( x = \frac{3}{2}y \) into Equation 1:
\[ \frac{3}{2}y + y = 40 \]
Combining the terms gives:
\[ \frac{5}{2}y = 40 \]

### Step 7: Solve for \( y \)
To find \( y \):
\[ y = 40 \cdot \frac{2}{5} = 16 \, kg \]

### Step 8: Solve for \( x \)
Now substitute \( y \) back into the equation for \( x \):
\[ x = \frac{3}{2} \cdot 16 = 24 \, kg \]

### Final Answer
The amounts of water taken from each tank are:
- \( x = 24 \, kg \) from tank A
- \( y = 16 \, kg \) from tank B

---