Fix dimensions of a conical shape water tank to store 2 lakh litre of water, having height twice the base dimensions.

To determine the dimensions of a conical water tank that can store 2 lakh liters (or 200,000 liters) of water, with the height being twice the radius of the base, follow these steps.

### Step 1: Convert volume to cubic meters
First, convert the volume from liters to cubic meters:
\[
200,000 \, \text{liters} = 200,000 \, \text{L} \times 0.001 \, \frac{\text{m}^3}{\text{L}} = 200 \, \text{m}^3
\]
The volume of the water tank is \(200 \, \text{m}^3\).

### Step 2: Formula for the volume of a cone
The volume \(V\) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where:
- \(r\) is the radius of the base,
- \(h\) is the height of the cone.

We know the height is twice the radius, i.e., \(h = 2r\).

### Step 3: Substitute \(h = 2r\) into the volume formula
Substitute the height in terms of the radius into the volume formula:
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Now set the volume equal to 200 m³:
\[
\frac{2}{3} \pi r^3 = 200
\]

### Step 4: Solve for the radius \(r\)
To find \(r\), rearrange the equation:
\[
r^3 = \frac{200 \times 3}{2 \pi} = \frac{600}{2 \pi} = \frac{300}{\pi}
\]
\[
r^3 \approx \frac{300}{3.1416} \approx 95.49
\]
\[
r \approx \sqrt[3]{95.49} \approx 4.57 \, \text{m}
\]

### Step 5: Calculate the height \(h\)
Since \(h = 2r\), the height of the cone is:
\[
h = 2 \times 4.57 \approx 9.14 \, \text{m}
\]

### Final Dimensions
- Radius \(r \approx 4.57 \, \text{m}\)
- Height \(h \approx 9.14 \, \text{m}\)

These are the approximate dimensions of the conical water tank that can store 200,000 liters of water.

Would you like further clarification or a breakdown of the calculations? Here are 5 related questions to expand on this topic:

1. How would the dimensions change if the height were triple the radius?
2. What if the volume of the tank needed to be 500,000 liters instead?
3. How would you calculate the surface area of this conical tank?
4. What are the structural advantages of a conical tank over other shapes for storing liquids?
5. How does the relationship between height and base radius affect the stability of the tank?

**Tip**: Always check the unit conversions when working with large volumes to ensure consistency across calculations.

Learn how to calculate the dimensions of a conical water tank that stores 200,000 liters of water, with a height twice the radius of its base. This problem involves geometric concepts and the volume formula for a cone. Suitable for students in Grades 9-12.

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