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Here are a few recommended readings before getting started with this lesson.
Consider a hemisphere, a cone, and a cylinder, all of which have the same radius. Each solid can be dragged and rotated. Create new solids by combining the given ones.
A solid that is made up of more than one solid is called a composite solid. The individual solids can be combined either by adding or subtracting them from one another. For instance, a hemisphere can be combined with a cone to make something that resembles a snow cone, or it could be used to dig a bowl shape out of a cylinder.
Ramsha has recently learned how to find the volume of composite solids. She is curious about finding the volumes of composite solids that she encounters in her daily life. Consider the diagram of a traffic cone she passed during her walk to school.
The volume occupied by the traffic cone is the sum of the volume of the prism base and the volume of the cone part.
r=5, h=30
Calculate power
b1⋅a=ba
Use a calculator
Round to nearest integer
A double-walled glass cup is a special cup with two layers of glass that help keep the drink at the right temperature, whether hot or cold. Ramsha has one of these cups. Her cup is cylindrical with a radius of 4 centimeters and a height of 12 centimeters. The second wall of the cup creates a cone.
Ramsha fills the cup with water.
Ramsha will fill the cone with water, so the volume of the cone is needed. The cone has the same height and radius as the cylinder, measuring 12 centimeters and 4 centimeters, respectively.
h=12, r=4
Calculate power
Multiply
Commutative Property of Multiplication
b1⋅a=ba
Calculate quotient
Next, Ramsha needs to find the volume of the air between the two walls of the cup. The first step is to find the volume of the shell of the cup. The cup has a cylindrical shape with a height of 12 centimeters and a radius of 4 centimeters.
Cancel out common factors
Simplify quotient
ba=b/64a/64
ba=a÷b
Convert to percent
Round to 1 decimal place(s)
Ramsha also wants to calculate the surface area of her double-walled glass cup.
The surface area of the cup consists of the lateral area of the cylinder, one of the bases of the cylinder, and the lateral area of the cone.
The double-walled glass cup is made up of a cylinder with a cone inside. To calculate its surface area, the lateral areas of the cylinder and the cone, along with one base area of the cylinder, need to be calculated.
Notice that only one end of the cylinder is closed, so only the sum of the lateral area and the base area of the cylinder will be calculated.
h=12, r=4
r=4
Calculate power
Commutative Property of Multiplication
Substitute values
Calculate power
Add terms
LHS=RHS
Rearrange equation
ℓ=160, r=4
Commutative Property of Multiplication
Ramsha bought a pencil with a radius of 3 millimeters. The total length of the pencil, excluding the eraser, is 160 millimeters. Moreover, the tip of the pencil is a 10-millimeter high cone.
The pencil is made of a cone, a cylinder, and a hemisphere, all with the same radius. The volume of the pencil equals the sum of the volumes of each solid.
Volume of a Cone | Volume of a Cylinder | Volume of a Hemisphere |
---|---|---|
V1=31πr2h | V2=πr2h | V3=32πr3 |
Use a calculator to make the calculations easier.
r=3, h=10
Calculate power
Multiply
Commutative Property of Multiplication
b1⋅a=ba
Calculate quotient