| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| {{ 'ml-lesson-time-estimation' | message }} |
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.
The volume of a composite solid is either the sum or the difference between the volumes of the individual solids, whichever is applicable. The surface area of a composite solid is the sum of the faces that enclose the solid.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 height of the cone part is 30 inches and its radius is 5 inches. The prism below the cone is a square prism with side lengths of 14 inches and a height of 1 inch. Help Ramsha find the volume of the traffic cone. Use a calculator for calculations and round the result to the nearest whole number.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.
The volume of a cone is one third the product of π, the square of the radius, and the height.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.
The volume of a cylinder is the product of π, the square of the radius, and the height.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.
Calculate the surface area of the cup with her. Round the answer to two decimal places.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.
The lateral area of a cylinder is twice the product of π, the radius, and the height.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.
Assuming the eraser is half of a sphere, what is the volume of the pencil? Round the answer to one decimal place.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
r=3
Calculate power
Commutative Property of Multiplication
ca⋅b=ca⋅b
Multiply
Calculate quotient
Substitute values
Factor out π
Add terms
Commutative Property of Multiplication
Use a calculator
Round to 1 decimal place(s)
Ramsha's house is a rough composite solid consisting of a square pyramid with a height of 8 feet and a base side length of 30 feet on top of a square prism.
Ramsha's father decides to cover the roof of their house with waterproof insulation material. Help Ramsha and her father calculate how many square feet of insulation material are needed.Ramsha only needs to know the lateral area of the pyramid. Use the Pythagorean Theorem to find the slant height.