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    • 2015 Publication >
      • Historical and Current Explanations >
        • Bell Curves
        • Birds Vs. Turbines
        • Energy in the Obama Era
        • The Future of Neuroscience
        • Gender Gap in Math
        • GMOs--Yes or No?
        • The History of Minecraft: How a Swedish Indie Game Came to Dominate the World
        • The Effect of Prozac on the Brain
        • Philae Lander's Discovery of Organic Molecules
        • Advantages and Disadvantages of Wind Turbines
        • Your Own Worst Enemy: An Overview of Lupus
        • The Methylhex Ban
        • The Effect of Lyme Disease on the Immune system
        • Infectious Mononucleosis
        • Replacing CFCs
        • The Switch
      • Mathematic and Scientific Explorations >
        • The 43rd Figure
        • The Clock
        • The Collatz Conjecture
        • Constructing a Soccer Ball
        • Determining how Ballparks Affect Batter's Ability to Create Hits
        • The Rotating Conundrum
        • Pythagorean Puzzle
        • Mathematic and Scientific Explorations
        • Kinetics Lab
        • Math in the Restaurant Business
        • Math as a Vessel for Social Change
        • Sustainability of Bottled Vs. Tap Water
        • Thoughts on the Lottery
        • Understanding Player Efficiency Rating
      • Scientific Research >
        • Communicating With Computers
        • The Mystery of Asthma
        • The Nanoscopic War Against Cancer
        • Phytochemistry
        • Solving the energy crisis with Intermediate Band Solar Cells
        • A Pain That Never Ends
        • Rapamycin Resistance
        • Ampacity of a Single Core Horizontal Cable
        • Morphological Properties of Texting Acronym Formation
        • cGAS and STING Expression
      • Reactions and Responses >
        • Can Humans Survive the Climate Crisis?
        • My Experience as a Teacher's Assistant
        • Ted Talk Responses
        • Teens For Food Justice
      • Artistic Creations >
        • Chandelier
        • Deltoidal Hexacontrahedon
        • Dodecahedron Card Trick
        • Eye of the Triangle
        • Free Radric Delantic Davis
        • The Grid
        • What Does A Randomly Composed Song Sound Like?
        • Science Wing Mural
    • 2014 Publication >
      • Cover Photo
      • Artistic Creations >
        • Art Using the Fibonacci Sequence
        • Computer Generated Architecture and Designs
        • Mathematical Landscape
        • Math Art
        • Math in Music
      • Historical and Current Explanations >
        • Algae Bio-Fuel
        • An Energy Alternative
        • Clean Energy In Transportation
        • Calorie Restriction
        • Creating Energy in the Modern World
        • Dietary Intervention Impact on Gut Microbial Gene Richness
        • Earthly Applications for NASA Technology
        • Explaining Relative Motion
        • Exploring Artificial Inteligence
        • Gamma Function
        • How Leaves Work
        • Hydrogen Fuel Cells
        • Music and Brain Development
        • Programming Calculators
        • The Science of Microsatellites
        • Sci-Fi Taser
        • Sloane's Gap
        • Sustainable Energy: Why Some Ideas Shine Brighter than Others
        • Understanding The Galvanic Cell
        • The Virus: Our Unforeseen Philosopher's Stone
        • What Are Fuel Cells and How Do They Work?
      • Mathematic and Scientific Explorations >
        • Astrocytes Expressing ALS-Linked Mutated SOD1 Release Factors Selectively Toxic to Motor Neurons
        • Big Bang
        • Dictyostelium Discoideum
        • The Future of Solar Cell Technology
        • And Many More...
      • Reactions and Responses >
        • Alternative Energy Sources, New but Unused
        • An Insight Into the Curious World of Ethnobotany
        • Challenging What We Think We Know
        • The Current State of American Education
        • Discovering New Numbers
        • Interview With an Architect
        • Life of Pi Response
        • Mathematical Art Video Commentary
        • Missing from Science Class
        • The Museum of Math
        • The Inside Scoop on a Real Mathematician
    • 2013 Publication

HOW MUCH DIRT DOES X-RAY DIG A DAY?

Editor's Note: Three Lower School students, Sam, Sam, and Sophie, worked with teacher Christopher Natalie to solve a problem concerning the the volume of a cylinder, without ever having heard of pi or knowing the equation for the volume of a cylinder! 
What’s The Problem?
Stanley’s shovel is five feet long. In the story, the character Mr. Sir tells the boys “The shovel is your measuring stick.”  He also tells them that the hole has to be five feet in diameter and five feet deep.  The shovel that the character X-Ray used was shorter than all the rest, but we didn’t know how much shorter.  We imagined that X-Ray’s shovel was four feet and eight inches long, so he makes a smaller hole.  But how much smaller?????

We started to think....


We needed to know how to find the volume of a cylinder, but didn’t know how.   We discussed how to find the volume of a cube and to apply that knowledge to a cylinder.

Cubes Helping Us Think About Cylinders

We know that we can find the volume of a cube by finding the area of a square and then multiplying it by the height of the cube.  So, we knew that we had to find the area of a circle.  How, though?

When you add sides to a square, one at a time, it gets more and more “circle-like”. 

We played around with cutting a circle into a bunch of triangles.  They weren’t triangles at first, though, because the base of the shape was round.  It looked more like a slice of pizza, not a triangle. But, when we cut it into more and more triangles, the base became a straighter line.  

We then cut the triangles apart and rearranged them so that they made a
Picture

Picture

We knew that we could cut one of the triangles in half, move one of those halves to the other side and it would become a rectangle.  The area of a rectangle is found by multiplying the length times the width.

See below:

Parallelogram:                                                                 Rectangle:
Picture
How did we use this information to find the area of a circle?
One of the sides of that rectangle, was really the circumference in disguise.  The other side was the radius in disguise.  That meant, if we could find the circumference of the circle and multiply it by the radius, we’d have the area of the circle.

Finding the radius of a circle

This was the easy part.  The radius is half of the diameter.  We knew that the diameter was 60” (5 foot long shovel).  Half of that is 30 inches.

Finding the circumference...
We explored the relationship between the circumference of a bunch of different circles and their diameters.  We realized that, when you take a piece of string as long as the diameter, and wrap it around the circumference of that circle, it always goes 3 times, with a little more left over.  We discussed this with Mr. Natale and learned that this relationship is known as pi (π).  For any circle, you can wrap the diameter around the circumference about 3.14 times.
Picture
Each colored line around the edge is a diameter length

We needed to find the circumference of Stanley’s hole.  So we multiplied the circumference by pi...

603.14=188.4. 

Finding the area of the circle
Then to find the area we had to do radius x1/2 circumference. The radius was 30 inches. Half the circumference is 94.2. So we did 30x 94.2 =2826!

Using the area of the circle to find the volume of the cylinder
To find the volume we multiplied the area of the circle by the height of the hole [60in.] and 60 x 2,826 = 169,560sqin. The answer of how much dirt Stanley digs in day!

How much less does X-Ray dig?
We did the same thing for X-Rays hole. The diameter was 56in. 56 x 3.11 = 175.84. 175.84 is the circumference. Half is 87.92. To find the area we multiplied the radius [28in] by 87.92  which equals 2,461.76sqin. We did the area x the diameter which equals 137,858.56sqin. And that’s how much dirt X-ray digs a day.

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  • Home
  • Who We Are
  • HOW TO SUBMIT
  • Past Publications
    • 2019 Publication >
      • Scientific Research
      • Mathematical Exploration
      • Scientific Exploration
      • Computer Science
    • 2018 Publication >
      • Artistic Creations
      • Historical and Current Explanations
      • Mathematic and Scientific Exploration
      • Scientific Research
    • 2017 Publication >
      • Artistic Creations
      • Historical and Current Explanations
      • Mathematic and Scientific Exploration
      • Reactions and Responses
      • Scientific Research
    • 2016 Publication >
      • Historical and Current Explanations
      • Mathematic and Scientific Explorations
      • Scientific Research
      • Reactions and Responses
      • Artistic Creations
    • 2015 Publication >
      • Historical and Current Explanations >
        • Bell Curves
        • Birds Vs. Turbines
        • Energy in the Obama Era
        • The Future of Neuroscience
        • Gender Gap in Math
        • GMOs--Yes or No?
        • The History of Minecraft: How a Swedish Indie Game Came to Dominate the World
        • The Effect of Prozac on the Brain
        • Philae Lander's Discovery of Organic Molecules
        • Advantages and Disadvantages of Wind Turbines
        • Your Own Worst Enemy: An Overview of Lupus
        • The Methylhex Ban
        • The Effect of Lyme Disease on the Immune system
        • Infectious Mononucleosis
        • Replacing CFCs
        • The Switch
      • Mathematic and Scientific Explorations >
        • The 43rd Figure
        • The Clock
        • The Collatz Conjecture
        • Constructing a Soccer Ball
        • Determining how Ballparks Affect Batter's Ability to Create Hits
        • The Rotating Conundrum
        • Pythagorean Puzzle
        • Mathematic and Scientific Explorations
        • Kinetics Lab
        • Math in the Restaurant Business
        • Math as a Vessel for Social Change
        • Sustainability of Bottled Vs. Tap Water
        • Thoughts on the Lottery
        • Understanding Player Efficiency Rating
      • Scientific Research >
        • Communicating With Computers
        • The Mystery of Asthma
        • The Nanoscopic War Against Cancer
        • Phytochemistry
        • Solving the energy crisis with Intermediate Band Solar Cells
        • A Pain That Never Ends
        • Rapamycin Resistance
        • Ampacity of a Single Core Horizontal Cable
        • Morphological Properties of Texting Acronym Formation
        • cGAS and STING Expression
      • Reactions and Responses >
        • Can Humans Survive the Climate Crisis?
        • My Experience as a Teacher's Assistant
        • Ted Talk Responses
        • Teens For Food Justice
      • Artistic Creations >
        • Chandelier
        • Deltoidal Hexacontrahedon
        • Dodecahedron Card Trick
        • Eye of the Triangle
        • Free Radric Delantic Davis
        • The Grid
        • What Does A Randomly Composed Song Sound Like?
        • Science Wing Mural
    • 2014 Publication >
      • Cover Photo
      • Artistic Creations >
        • Art Using the Fibonacci Sequence
        • Computer Generated Architecture and Designs
        • Mathematical Landscape
        • Math Art
        • Math in Music
      • Historical and Current Explanations >
        • Algae Bio-Fuel
        • An Energy Alternative
        • Clean Energy In Transportation
        • Calorie Restriction
        • Creating Energy in the Modern World
        • Dietary Intervention Impact on Gut Microbial Gene Richness
        • Earthly Applications for NASA Technology
        • Explaining Relative Motion
        • Exploring Artificial Inteligence
        • Gamma Function
        • How Leaves Work
        • Hydrogen Fuel Cells
        • Music and Brain Development
        • Programming Calculators
        • The Science of Microsatellites
        • Sci-Fi Taser
        • Sloane's Gap
        • Sustainable Energy: Why Some Ideas Shine Brighter than Others
        • Understanding The Galvanic Cell
        • The Virus: Our Unforeseen Philosopher's Stone
        • What Are Fuel Cells and How Do They Work?
      • Mathematic and Scientific Explorations >
        • Astrocytes Expressing ALS-Linked Mutated SOD1 Release Factors Selectively Toxic to Motor Neurons
        • Big Bang
        • Dictyostelium Discoideum
        • The Future of Solar Cell Technology
        • And Many More...
      • Reactions and Responses >
        • Alternative Energy Sources, New but Unused
        • An Insight Into the Curious World of Ethnobotany
        • Challenging What We Think We Know
        • The Current State of American Education
        • Discovering New Numbers
        • Interview With an Architect
        • Life of Pi Response
        • Mathematical Art Video Commentary
        • Missing from Science Class
        • The Museum of Math
        • The Inside Scoop on a Real Mathematician
    • 2013 Publication