The area of rectangular based tanlē "of which longer side is 150% more than smaller side is `1440 m^(2)` and the tank contains `10800 m^(3)` water.
If radius of a conical tank is `(7)/(8)th` of smaller side of rectangular based tank and height is two times of height of rectangular based tank, then find capacity of water contained by conical tank?

Options

A. 6730 `m^(3)`
B. 6530 `m^(3)`
C. 6930 `m^(3)`
D. 6960 `m^(3)`

Answer

To solve the problem step by step, we will follow the instructions given in the question and use the information provided.

### Step 1: Define the dimensions of the rectangular tank
Let the smaller side of the rectangular tank be \( x \) meters. According to the problem, the longer side is 150% more than the smaller side. Therefore, the longer side can be expressed as:
\[ \text{Longer side} = x + 1.5x = 2.5x \]

### Step 2: Calculate the area of the rectangular tank
The area of the rectangular tank is given as \( 1440 \, m^2 \). The area can also be calculated using the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} = (2.5x) \times x = 2.5x^2 \]
Setting this equal to the given area:
\[ 2.5x^2 = 1440 \]

### Step 3: Solve for \( x \)
To find \( x \), we rearrange the equation:
\[ x^2 = \frac{1440}{2.5} = 576 \]
Taking the square root of both sides:
\[ x = \sqrt{576} = 24 \, \text{meters} \]

### Step 4: Calculate the dimensions of the rectangular tank
Now that we have \( x \):
- Smaller side (breadth) = \( 24 \, m \)
- Longer side (length) = \( 2.5 \times 24 = 60 \, m \)

### Step 5: Find the height of the rectangular tank
The volume of the rectangular tank is given as \( 10800 \, m^3 \). The volume can be calculated using the formula:
\[ \text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height} \]
Substituting the known values:
\[ 10800 = 60 \times 24 \times h \]
Calculating \( 60 \times 24 \):
\[ 10800 = 1440h \]
Now, solving for \( h \):
\[ h = \frac{10800}{1440} = 7.5 \, m \]

### Step 6: Calculate the dimensions of the conical tank
The radius of the conical tank is given as \( \frac{7}{8} \) of the smaller side of the rectangular tank:
\[ \text{Radius} = \frac{7}{8} \times 24 = 21 \, m \]
The height of the conical tank is twice the height of the rectangular tank:
\[ \text{Height of conical tank} = 2 \times 7.5 = 15 \, m \]

### Step 7: Calculate the capacity of the conical tank
The capacity of the conical tank can be calculated using the formula for the volume of a cone:
\[ \text{Volume} = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times (21)^2 \times 15 \]
Calculating \( (21)^2 = 441 \):
\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times 441 \times 15 \]
Calculating \( \frac{22 \times 441 \times 15}{3 \times 7} \):
\[ = \frac{22 \times 441 \times 15}{21} = 22 \times 21 \times 15 = 6930 \, m^3 \]

### Final Answer
The capacity of water contained by the conical tank is \( 6930 \, m^3 \).

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