Basic Math for the Small Public Water Systems Operator

1
Basic Math for the Small Public Water Systems
Operator

• Small Public Water Systems
  Technology Assistance Center Penn State
  Harrisburg

2
Math Training Module Two

• Volume
• In this module we will expand on our knowledge of
  calculating area and learn to calculate the
  volume of the various shaped containers that are
  components of a water treatment system.

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Basic Three Dimensional Shapes

• Rectangle
• Cylinder
• Pyramid or Prism
• Cone ? Sphere

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Units of Measure - Volume

• Volumes are measured in cubic units.
• The English system of volume measurement
  includes the following units
• Cubic Inches Cubic Yards
• Cubic Feet Gallons
• The Metric system of volume measurement includes
  the following units
• Cubic Centimeters Cubic Meters
• Liters

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• Volume calculations are three dimensional. They
  involve three dimensions such as length, width
  and depth.
• For example when we multiply the linear unit feet
  times the linear unit feet times the linear unit
  feet we get the volume unit measurement of cubic
  feet.
• So the unit multiplication ft x ft x ft gives the
  answer ft3 or cu ft.

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• An example in the Metric system of measurement
  would be to multiply the linear unit meter times
  the linear unit meter times the linear unit meter
  for a result of m3 or cu m.

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Calculating the Volume of a Rectangular Vessel

• The formula to calculate the volume of a
  rectangle is
• Volume Length x Width x Depth
• Volume (l) x (w) x (d)

Depth
Length
Width
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Example

• Calculate the volume of a rectangular tank with a
  length of 25 feet, a width of 10 feet and a depth
  of 15 feet.
• Volume (l) x (w) x (d)
• Volume 25 ft x 10 ft x 15 ft
• Volume 3,750 cubic feet

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Practice Exercise

• 1. Calculate the volume of a rectangular tank
  with a length of 50 feet, a width of 25 feet and
  a depth of 15 feet.

Answer 18,750 cu ft
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Solution

• Volume Length x Width x Depth
• Volume 50 ft x 25 ft x 15 ft
• Volume 18,750 ft3

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Practice Exercise

• 2. Calculate the volume of a rectangular tank
  with a length of 30 meters, a width of 10 meters
  and a depth of 10 meters.

Answer 3,000 cu m
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Solution

• Volume Length x Width x Depth
• Volume 30 m x 10 m x 10 m
• Volume 3,000 m3

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Calculating the Volume of a Cylinder

• The formula to calculate the volume of a cylinder
  is
• Volume Area of Circle x Depth
• or
• Volume ? x r2 x d
• ? 3.14

Radius
Depth
14
Example

• Calculate the volume of a cylinder with a radius
  of 5 feet and a depth of 15 feet.
• Volume ? x r2 x d
• Volume 3.14 x (5 feet)2 x 15 feet
• Volume 3.14 x 25 ft x 15 ft
• Volume 1,178 cu ft

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Practice Exercise

• 1. Calculate the volume of a cylindrical storage
  tank with a radius of 10 feet and a depth of 30
  feet.

Answer 9,420 cu ft
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Solution

• Volume ? x r2 x d
• Volume ? x (10 ft)2 x 30 ft
• Volume 9,420 ft3

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Practice Exercise

• 2. Calculate the volume of a cylindrical storage
  tank with a diameter of 10 feet and a depth of 30
  feet.

Answer 2,355 cu ft
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Solution

• Volume ? x r2 x d
• Volume ? x (5 ft)2 x 30 ft
• Volume 2,355 ft3

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Calculating the Volume of a Cone

• The formula for calculating the area of a cone
  is
• Volume ? x r2 x depth
• 3
• Notice The volume of a cone is one-third the
  volume of a cylinder.

Radius
Depth
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Example

• Calculate the volume of a cone with a radius of
  10 feet and a depth of 15 feet.
• Volume ? x r2 x depth
• 3
• Volume 3.14 x (10 ft)2 x 15 ft
• 3
• Volume 1,570 cu ft

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Practice Exercise

• 1. Calculate the volume of cone with a radius of
  5 feet and a depth of 5 feet.

Answer 131 cu ft
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Solution

• Volume ? x r2 x depth
• 3
• Volume ? x (5 ft)2 x 5 ft
• 3
• Volume 131 ft3

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Practice Exercise

• 2. Calculate the volume of cone with a diameter
  of 5 feet and a depth of 10 feet.

Answer 65 cu ft
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Solution

• Volume ? x r2 x depth
• 3
• Volume ? x (2.5 ft)2 x 10 ft
• 3
• Volume 65 ft3

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Calculating the Volume of a Triangle

• The formula to calculate the volume of a
  triangular vessel or a trough is
• Volume Area of Triangle x Length of Trough
• or
• Volume base x height x length
• 2

Height
Length
Base
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Example

• Calculate the volume of a triangle with a base of
  8 feet, a height of 5 feet and a length of 8
  feet.
• Volume Base x Height x Length
• 2
• Volume 8 ft x 5 ft x 8 ft
• 2
• Volume 160 cu ft

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Practice Exercise

• 1. Calculate the volume of a triangle with a
  base of 15 feet, a height of 10 feet and a length
  of 12 feet.

Answer 900 cu ft
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Solution

• Volume Base x Height x Length
• 2
• Volume 15 ft x 10 ft x 12 ft
• 2
• Volume 900 ft3

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Practice Exercise

• 2. Calculate the volume of a triangle with a
  base of 20 feet, a height of 15 feet and a length
  of 10 feet.

Answer 1,500 cu ft
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Solution

• Volume base x height x length
• 2
• Volume 20 ft x 15 ft x 10 ft
• 2
• Volume 1,500 ft3

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Calculating the Volume of a Sphere

• The formula to calculate the volume of a sphere
  is
• Volume ? x (diameter)3
• 6
• Where ? 3.14

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Example

• Calculate the volume of a sphere with a diameter
  of 15 feet.
• Volume 3.14 x (15 ft)3
• 6
• Volume 1,767 cu ft

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Practice Exercise

• 1. Calculate the volume of sphere with a
  diameter of 20 feet.

Answer 4,187 cu ft
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Solution

• Volume ? x (diameter)3
• 6
• Volume 3.14 x (20 ft)3
• 6
• Volume 4,187 ft3

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Practice Exercise

• 2. Calculate the volume of sphere with a
  diameter of 12.5 feet.

Answer 1,022 cu ft
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Solution

• Volume ? x (diameter)3
• 6
• Volume 3.14 x (12.5 ft)3
• 6
• Volume 1,022 ft3

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Calculating the Surface Area of a Sphere

• The formula for calculating the surface area of a
  sphere is denoted by A
• As ? x D2

s
Diameter
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Example

• Calculate the surface area of a sphere with a
  diameter of 8 feet.

As ? x D2
As 3.14 x (8 ft)2
As 201 ft2
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Practice Exercise

• 1. Calculate the surface area of a sphere with a
  diameter of 3 feet.

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Solution

• As ? x D2
• As 3.14 x (3 ft)2
• As 28 ft2

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Practice Exercise

• 2. Calculate the surface area of a sphere with a
  diameter of 6 feet.

Answer 113 ft2
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Solution

• As ? x D2
• As 3.14 x (6 ft)2
• As 113 ft2

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Conclusion

• This concludes the math module for calculating
  volumes.
• If you need more practice, run the slide show
  again.
• If you feel confident about calculating volumes,
  try some of our other math modules.