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Many objects used in daily life can be modeled as three-dimensional figures. In this lesson, geometric figures will be used to describe some daily life objects.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Comparing Aquarium Volumes
Dominika wants to buy a new aquarium for her fish. She is interested in two types of aquariums that have equal linear measurements. 
Aquarium is a right cylinder with a diameter of feet and a height of feet. Additionally, a right cone sits inside of it. Aquarium is a hemisphere with a diameter of feet. Help Dominika answer the following questions.
a What is the volume of Aquarium If necessary, round the answer to the nearest cubic foot.
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b For each aquarium, what is the area of the water’s surface when filled to a height of feet? Write the answer in terms of and
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c Use the answers found above to find the volume of the other aquarium.
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Example
Comparing Aquarium Volumes II
Recall how the formula for the volume of a sphere is proven. The same thought process used in the proof can be applied to solve the challenge.
Two types of aquariums attract the attention of Dominica: Aquarium and Aquarium
a What is the volume of Aquarium Round the answer to the nearest cubic foot.
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b For each aquarium, what is the area of the water’s surface when filled to a height of feet? Write the answer in terms of and
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c Use the answers found above and find the volume of the other aquarium.
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Hint
a Subtract the volume of the cone from the volume of the cylinder.
b Examine how the cross-sections that are parallel to the bases change.
c What does the Cavalieri's Principle states?
Solution
a By subtracting the volume of the cone from the volume of the cylinder, the volume of the Aquarium can be found.
Recall the formulas for the volumes of a cylinder and a cone.
The difference between and will give the number of cubic feet of water that Aquarium can hold.
Aquarium can hold about cubic feet of water.
| Volume of the Cylinder | Volume of the Cone | |
|---|---|---|
| Formula |
Both the cone and the cylinder forming Aquarium have a foot diameter, and therefore both have a radius of feet. With that in mind, substitute and
| Volume of the Cylinder | Volume of the Cone | |
|---|---|---|
| Formula | ||
| Substitute Values | ||
| Calculate |
▼
Evaluate right-hand side
b Begin by examining the cross-sections of each aquarium. Then, write an expression for the area of the water’s surface at height 
With this in mind, consider the vertical and horizontal cross-sections of the aquarium.
As seen on the diagram, the vertical cross-sections of the water — the shaded triangles — are isosceles right triangles, and the horizontal cross-sections form two concentric circles. Using the horizontal cross-sections, the area of the water’s surface at height will be found. To do so, subtract the area of the inner circle from the area of the outer circle
When Aquarium is filled to a height of feet, the area of the water’s surface is square feet. 
The diagram shows a right triangle Its hypotenuse and length of one of its legs is known in terms of Therefore, using the Pythagorean Theorem, can also be expressed in terms of
Now, considering the horizontal cross-sections of Aquarium will be the radius of the inner circle. 
In this instance, the area of the water’s surface at height is the area of a circle with a radius of
For Aquarium the area of the water’s surface at height is the same as the other aquarium, square feet.
Aquarium
Since the cone inside the aquarium is a right cone, its vertex is directly above the center of its base. Furthermore, its height and radius have the same length. Therefore, an isosceles right triangle inside the cone can be formed, as indicated in the diagram.SubstituteValues
Substitute values
▼
Evaluate
CalcPow
Calculate power
CommutativePropMult
Commutative Property of Multiplication
Distr
Distribute
SubTerm
Subtract term
Aquarium
Examine the vertical cross-sections of the hemisphere. c Recall what the Cavalieri Principle states.
According to the Cavalieri Principle, the volumes of water must be equal when both aquariums are filled. Therefore, no further calculations are necessary — Aquarium will hold about cubic feet of water.
|
Cavalieri Principle |
|
Two solids with the same height and the same cross-sectional area at every altitude have the same volume. |
Both aquariums have a height of feet, and the area of the water’s surface when filled to a height of feet is the same for each aquarium.
| Aquarium $\bm{A}$ | Aquarium $\bm{B}$ | |
|---|---|---|
| Height | ||
| Cross-Sectional Area | ||
Example
Finding the Length of a Toilet Paper Roll
Tiffaniqua wants to calculate the length of the a toilet paper roll. Hey! It is on a great sale, Okay. She draws a diagram and denotes the thickness of the paper, the inner radius, and the outer radius by and respectively.
a Write an expression for the length of the paper roll in terms of and
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b Find the length of the paper roll if and all measured in centimeters.
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Solution
a Consider the following horizontal cross-section of the toilet roll. The cross-section consists of concentric circles.
The area of the shaded region can be calculated in two ways. It can be expressed as the area of the circle of radius minus the area of the circle of radius
Alternatively, it can be expressed as the area of the front face of the long thin rectangular prism, which is created when the paper is unrolled. 
The area of the front face is the product of and That product is also equal to Therefore, can be substituted for in the derived formula for the area of the shaded region.
Note that there might be several alternative ways to find the equation. Only one possible way was shown here.
b By substituting and into the previously derived equation, the length of the paper towel will be calculated.
The length of the paper is about centimeters.
Example
Cylindrical Soda Can
A cylindrical soda can is made of aluminum. It is inches high and its bases have a radius of approximately inches.
Give a go at answering the following set of questions. If necessary, round the answer to two decimal places.
a Find the surface area of the soda can.
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b The density of aluminum is approximately grams per cubic centimeter. If the mass of the soda can is approximately grams, how many cubic centimeters of aluminum does it contain?
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c Suppose that the thickness of the soda can is uniform throughout its body. Estimate the soda can's thickness.
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Hint
a Use the formula for the surface area of a cylinder.
b The density of a substance is equal to its mass divided by its volume.
c How might the surface area and volume of the soda can be related?
Solution
a Since the soda can is a cylinder, the formula for the surface area of a cylinder will be used.
Here, and are the radius and the height of the cylinder, respectively. The radius is inches and the height is inches. Substitute these values in the formula and evaluate it.
Therefore, the surface area of the soda can is about square inches.
b Recall that the density of a substance is equal to its mass divided by its volume. In other words, the volume of a substance is its mass divided by its density.
Since the mass of the aluminum can and the density of aluminum are given, the volume can be calculated. Substitute and into the equation.
The soda can is made of about cubic centimeters of aluminum.
c To determine the thickness of the soda can, its surface area and volume are good to use.
Let denote the thickness. Then, the volume of the aluminum part of the soda can will be equal to the surface area of the soda can multiplied by its thickness. Therefore, to find the the aluminum part's volume should be divided by the soda can's surface area.
However, since the amount of aluminum is found in cubic centimeters and the surface area of the soda can in square inches, a conversion factor must be used. Use to convert square inches to square centimeters.
Now, by substituting and can be found.
This means that the thickness of the soda can is about centimeters.
▼
Simplify
Example
Choosing Type of Glass
By modeling real-life objects using geometric shapes, various characteristics of the objects can be determined. These characteristics can then be compared to make inferences which could impact real decisions to be made.
Emily is attending a fair and wants to sell liters of homemade orange juice she is naming Oranjya Thirsty. She needs to decide the type of glass she will use to serve the juice — a cocktail glass or a Collins glass.
A cocktail glass is a type of glass that has an inverted cone bowl. The cone bowl's height is centimeters and the radius of its base is centimeters. A collins glass is a cylindrical glass with a height of centimeters and a radius of centimeters. Help Emily make a decision by answering the following questions.
a How many cocktail glasses of orange juice can she sell?
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b How many Collins glasses of orange juice can she sell?
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c If Emily chooses cocktail glasses, she will sell each for dollars. If the Collins glasses are chosen, each will be sold for dollars. Which glass type selection makes Emily more money?
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Hint
a Use the formula for the volume of a cone to determine how many liters of orange juice a cocktail glass can hold.
b Use the formula for the volume of a cylinder to determine how many liters of orange juice a Collins glass can hold.
c Use the answers found in the previous parts.
Solution
a First, the volume of the bowl of a cocktail glass will be calculated. Then, it will be used to find the number of glasses. The bowl of a cocktail glass can be modeled by a cone as shown.
The height of the cone bowl and the radius of its base are and centimeters, respectively. Substitute these values into the volume formula of a cone.
This means that a cocktail glass can hold approximately cubic centimeters orange juice. Conversion needs to be made. The conversion factor will convert cubic centimeters into liters.
Finally, the number of cocktail glasses can be calculated.
Therefore, liters of orange juice fully fills cocktail glasses.
b Similarly, start by finding the volume of the cylindrical glass.
Its height is centimeters and its base radius is centimeters. Substitute these values into the formula.
This means that the volume of a collins glass is about cubic centimeters. After converting its unit of measurement into liters, it is liters. Now, the volume of a carton of orange juice can be divided by the volume of a collins glass. Therefore, liters of orange juice fully fills collins glasses.
c Emily can sell cocktail glasses or collins glasses. Knowing the prices, the revenue from each sale can be calculated.
As a result, Emily should choose the Collins glass, as the revenue from this selection is greater.
| Type of Glass | |
|---|---|
| Cocktail Glass | Collins Glass |
Example
Estimating How Many Grains of Sand a Hand Can Hold
With the help of geometric modeling, any number of objects can be approximated regardless of whether they are super large or tiny minuscule grains of sand.
Take, for example, Ramsha's situation. She is looking through photos from her trip to the beach to post on her social media page. A photo that shows her holding sand sparks her curiosity. She wonders how many individual grains of sand is she holding. Ramsha thinks she can model a grain of sand using a sphere. She then assumes that each grain has a diameter of centimeters.
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Ramsha figures she can hold grams of sand in her hands. If the density of sand is approximately grams per cubic centimeter, help Ramsha approximate the number of grains of sand in her hands. Write the answer in scientific notation.
Answer
About
Hint
The formula for the volume of a sphere is where is the radius of the sphere.
Solution
To find the number of grains of sand in Ramsha's hands, the mass of the sand in her hands should be divided by the mass of a grain.
Recall that the density of a substance is equal to its mass divided by its volume. In other words, the mass of a substance is its density times its volume.
Since the density of a grain of sand is given, the volume of a grain of sand will be calculated first.
The volume of a grain is about cubic centimeters.
The number of grains of sand is approximately or about 1.7 million.
Finding the Volume of a Grain
The radius of a grain is centimeters. Use the formula for the volume of a sphere to find the volume of a grain.▼
Evaluate right-hand side
CalcPow
Calculate power
CommutativePropMult
Commutative Property of Multiplication
UseCalc
Use a calculator
RoundDec
Round to decimal place(s)
Write in scientific notation
Finding the Mass of a Grain
The density of a grain is and its volume is cubic centimeters. By multiplying these values, the mass of a grain can be found.▼
Evaluate
CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
CommutativePropMult
Commutative Property of Multiplication
Multiply
Multiply
RoundDec
Round to decimal place(s)
Finding the Number of Grains
Finally, substitute the values into the formula mentioned at the beginning to calculate the number of grains of sand.▼
Evaluate right-hand side
CrossCommonFac
Cross out common factors
SimpQuot
Simplify quotient
WriteProdFrac
Write as a product of fractions
CalcQuot
Calculate quotient
Multiply
Multiply
RoundSigDig
Round to significant digit(s)
Write in scientific notation
the total number of stars in the universe is greater than all the grains of sand on all the beaches of the planet Earth.
Closure
Modeling the Human Eye
Research projects usually require an interdisciplinary approach. That is, people from different disciplines work together to develop and test hypothesis, run experiments, and test theories.
Biologists, for example, can work with mathematicians to model a part of an organism. By doing so, researchers can predict how these parts function, grow, and change. For example, the human eye was able to be modeled as a sphere. Move the slider to rotate the eye.
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These interdisciplinary studies may sometimes result in new discoveries, such as the scutoid, a new geometric solid introduced in
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