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 Context

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.

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Key Insights

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:

Continue Reading

We compute $ y + \frac{1}{y} $. First, recall that $ x + \frac{1}{x} = 5 $. Square both sides:
x^2 + 2 + \frac{1}{x^2} = 25 \Rightarrow x^2 + \frac{1}{x^2} = 23
Now compute $ x^2 + 1 $ and $ x^2 - 1 $:

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Final Thoughts

[
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 ) m². 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
]