13.3 volumes of spheres
DESCRIPTION
13.3 Volumes of spheres. By Kylie Bolton and Jessica Nguyen. Objectives. Find volumes of spheres and hemispheres. Solve problems involving volumes of spheres and hemispheres with given diameters, radius’, surface areas, and circumferences. What is a Sphere?. - PowerPoint PPT PresentationTRANSCRIPT
13.3 VOLUMES OF SPHERES
By Kylie Bolton and Jessica Nguyen
Objectives Find volumes of spheres and
hemispheres. Solve problems involving volumes of
spheres and hemispheres with given diameters, radius’, surface areas, and circumferences.
What is a Sphere? A sphere is the set
of points in space equidistant from a certain point.
It has no faces, edges, or vertices.
An example is our earth.
How do we get the equation for the volume of a sphere?Finding the volume of a
sphere is kind of like finding the volume of a pyramid and the surface area of a sphere.
Suppose that the space inside a sphere is divided into an infinite number of near pyramids with all their vertices on the center of the sphere. The height of the pyramid equals the radius of the sphere.
The sum of all the pyramids’ bases will equal the surface area of the entire sphere. The sum of all the pyramids’ volume is equal to the volume of the sphere.
Recall, V(pyramid)=1/3Bh, where B is the area of its base and h is its height. Also, the surface area of a sphere, 4∏r², is equal to B1 + B2 + B3 +…+ Bn
V=1/3B¹h¹ + 1/3B²h² + 1/3B³h³ +…+ 1/3Bnhn
=1/3B1r + 1/3B2r + 1/3B3r +…+ 1/3Bnr
=1/3r(B1 +B2 + B3 +…+ Bn)
=1/3r(4∏r²) =4/3∏r³
Sum of the volumes of all the pyramids.
Replace h with r.
Distributive Property
Replace B1+B2+B3+…+Bn With 4∏r²
Simplify
Volume of a Sphere
If a sphere has a volume of a V cubic units and a radius of r units, then V=4/3r³
Find the Volume of the sphere if C=24m Round to the nearest 10th
Find rC=2∏r24=2∏rR≈3.82m
Plug it in!V=4/3∏(3.82)³V=233.5m³
Example 2:
Billy was delivering spherical teapots. He accidently spilt the tea out of one. So he needs to know how much tea to put in the teapots.
Help him out by finding the volume of the spherical teapot if the radius=12in.
Round to the nearest 10th.
12
Continued
Find the volume of the jug.
V=4/312³ V=7238.2in³
Substitution Multiply
Volume of a Hemisphere
Hemisphere- half of a sphere.
V=1/2(4/3r³)
Example:Find the volume of this hemisphere with a radius of 2in.
2in
Continued1. Plug the radius into the formula V=1/2(4/3r³)2. V=1/2(4/32³)3. V=1/2(33.5)4. V=16.8
FIND THE VOLUME OF THE HEMISPHERE IF THE SURFACE AREA=10ROUND TO THE NEAREST 10TH
Example 1:
Continued T=1/2(4r²)+ r² 10=1/2(4r²)+ r² 10=3r² 3.333=r² 1.83=r V=1/2(4/3r³) V=1/2(4/31.83³) V=12.8 units²
Surface area of a sphere
Substitute Divide both sides by
3∏ Find the sq. root Volume of a sphere Substitute Use a calculator
AssignmentPg. 704 #3-7, 9-20, 23-24, 30-31
By Jessica and Kylie!