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Here are a few recommended readings before getting started with this lesson.
VSphere=34πr3
The volume of a sphere with radius r is four-thirds the product of pi and the radius cubed.
Cavalieri's Principle will be used to show that the formula for the volume of a sphere holds. For this purpose, consider a hemisphere and a right cylinder with a cone removed from its interior, each with the same radius and height.
Now, consider a plane that cuts the solids at a height x and is parallel to the bases of the solids.Draw a right triangle with height x, base y, and hypotenuse r. Here, x is the distance between the center of the base of the hemisphere and the center of the cross sectional circle, y is the radius of the cross sectional circle, and r is the radius of the hemisphere.
Using the Pythagorean Theorem, an expression for y can be found.Substitute values
a=33⋅a
ca⋅b=ca⋅b
Subtract fractions
Last weekend, Tearrik installed a spherical water tank for his house.
If the tank has a radius of 2.15 feet, what is the maximum amount of water it can hold? Round the answer to two decimal places.
Find the volume of the tank.
The volume of the Earth is approximately 259875159532 cubic miles.
Assuming the Earth is spherical, what is the Earth's radius? Round the answer to one decimal place.
Solve the volume formula of the sphere for the radius and substitute the given volume.
LHS⋅3=RHS⋅3
LHS/4π=RHS/4π
Rearrange equation
3LHS=3RHS
3a3=a
V=259875159532
Use a calculator
Round to 1 decimal place(s)
Find the volume of the bucket and the volume of each balloon. Then, divide the volume of the bucket by the volume of a balloon.
To determine the minimum number of balloons needed to fill up the bucket, first solve for the volume of the bucket and the volume of each balloon. To do so, recall the formulas to find the volume of a cylinder and a sphere.
Volume of a Cylinder | Volume of a Sphere |
---|---|
VC=πr2h | VS=34πr3 |
Cross out common factors
Cancel out common factors
b/ca=ba⋅c
Multiply
Calculate quotient
Ramsha bought a standard pencil whose radius is 4 millimeters and the length, not including the eraser, is 180 millimeters. After a good sharpening, the tip turned into a 12 millimeter high cone.
Assuming the eraser is half of a sphere, what is the volume of the pencil? Round the answer to two decimal places.
The pencil is made of a cone, a cylinder, and a hemisphere, all with the same radius. Therefore, 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 |
Consider the fact that baseballs commonly have a radius of 1.43 inches. With this information, the volume of a baseball can be calculated, right? What about the surface area? How much leather is needed to make the lining of a baseball?
To answer these question, the following formula can be used.
The surface area of a sphere with radius r is four times pi multiplied by the radius squared.
To find the surface area of a sphere, this informal justification will use the areas and volumes of known figures. It includes approximating the ratio between a sphere's surface area and its volume. In considering a known figure, suppose that a sphere is filled with n congruent pyramids. The area of the base of each pyramid is B.
The formula to find the volume of a pyramid uses the height and the area of its base. Here, since the base of the pyramid lies on the surface of the sphere, the height of the pyramid is equal to the radius r of the sphere.Substitute expressions
b/ca=ba⋅c
Cross out common factors
Cancel out common factors
Cancel out common factors
Simplify quotient
Substitute values