Math Problem Statement
Fix dimensions of a conical shape water tank to store 2 lakh litre of water, having height twice the base dimensions.
Solution
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: The volume of the water tank is .
Step 2: Formula for the volume of a cone
The volume of a cone is given by the formula: where:
- is the radius of the base,
- is the height of the cone.
We know the height is twice the radius, i.e., .
Step 3: Substitute into the volume formula
Substitute the height in terms of the radius into the volume formula: Now set the volume equal to 200 m³:
Step 4: Solve for the radius
To find , rearrange the equation:
Step 5: Calculate the height
Since , the height of the cone is:
Final Dimensions
- Radius
- Height
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:
- How would the dimensions change if the height were triple the radius?
- What if the volume of the tank needed to be 500,000 liters instead?
- How would you calculate the surface area of this conical tank?
- What are the structural advantages of a conical tank over other shapes for storing liquids?
- 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.
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Math Problem Analysis
Mathematical Concepts
Geometry
Volume of a Cone
Unit Conversion
Formulas
Volume of a cone: V = (1/3)πr²h
Height-radius relationship: h = 2r
Unit conversion: 1 L = 0.001 m³
Theorems
Volume Formula of a Cone
Suitable Grade Level
Grades 9-12
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