Question**: A cylindrical tank with a radius of 3 meters is filled with water to a height of 10 meters. If the water is transferred to a cuboid tank with a base area of 45 square meters, what will be the height of the water in the cuboid tank?
How to Calculate Water Height After Transfer from a Cylindrical to a Cuboidal Tank: A Practical Problem
When managing water storage systems, understanding how volume translates between different tank shapes is crucial. One common scenario involves transferring water from a cylindrical tank to a cuboid (box-shaped) tank. This article explains step-by-step how to calculate the water height in the cuboid tank when the original cylindrical tank holds water to a specified height.
Understanding the Problem
We begin with a cylindrical tank with:
- Radius ( r = 3 ) meters- Height of water ( h_{\ ext{cyl}} = 10 ) meters- Water transferred to a cuboid tank with base area ( A = 45 ) square meters
We need to find the final height of water ( h_{\ ext{cuboid}} ) in the cuboid tank.
Step 1: Calculate the Volume of Water in the Cylindrical Tank
The volume ( V ) of a cylinder is given by the formula:
[V = \pi r^2 h]
Substituting the known values:
[V = \pi (3)^2 (10) = \pi \ imes 9 \ imes 10 = 90\pi \ ext{ cubic meters}]
Using the approximation ( \pi pprox 3.1416 ):
[V pprox 90 \ imes 3.1416 = 282.74 \ ext{ cubic meters}]
Step 2: Relate Volume to Height in the Cuboid Tank
The cuboid tank has a base area (floor area) of ( 45 ) mÃÂò. The volume of water remains constant during transfer, so:
[V = \ ext{Base Area} \ imes \ ext{Height}]
[90\pi = 45 \ imes h_{\ ext{cuboid}}]
Solving for ( h_{\ ext{cuboid}} ):
[h_{\ ext{cuboid}} = rac{90\pi}{45} = 2\pi]
Using ( \pi pprox 3.1416 ):
[h_{\ ext{cuboid}} pprox 2 \ imes 3.1416 = 6.2832 \ ext{ meters}]
Final Answer
The height of the water in the cuboid tank will be ( 2\pi ) meters, or approximately 6.28 meters.
Why This Calculation Matters
Accurately determining water heights ensures proper infrastructure planning, prevents overflow, and supports efficient water management in residential, industrial, and agricultural systems. Understanding volume conservation across tank shapes helps engineers, architects, and facility managers optimize storage solutions.
Summary
- Cylinder volume: ( V = 90\pi , \ ext{m}^3 )- Cuboid base area: 45 mÃÂò- Water height in cuboid: ( h = rac{V}{\ ext{Area}} = rac{90\pi}{45} = 2\pi ) m- Approximate height: ~6.28 m
By mastering these calculations, you empower smarter design and management of water storage systems.
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