Understanding 2D and 3D Relationships: Volume and Modeling Guide
School
Canton High School, Canton, MI**We aren't endorsed by this school
Course
MATH 101
Subject
Mathematics
Date
Dec 10, 2024
Pages
12
Uploaded by HighnessPigeonMaster250
Unit 10 Assignment Sheet 3D and Modeling Date Lesson Homework Thurs May 11 2D and 3D Relationships Unit 10 Day 1 WS Fri May 12 Volume of Prisms Prism WS Mon May 15 Volume of Cylinders Cylinder WS Tues May 16 Quiz 1 Review Wed May 17 QUIZ 1 Thurs May 18 Volume of Pyramids Pyramid WS Fri May 19 Volume of ConesCone WS Mon May 22 Volume of Spheres Sphere WS Tues May 23 Quiz 2 Review Wed May 24 Quiz 2 Thurs May 25 Composite Solids Composite Solid WS Fri May 26 Modeling Unit 10 Review Mon May 29 No School Tue May 30 Unit 10 Review Day 2 Unit 10 Review Wed May 31 Unit 10 Test
Unit 10 –2D and 3D Relationships A is a solid that is bounded by polygons that enclose a single region of space. A baseis a polygon that is used to name the polyhedron. *There are always less ______________ than ______________. The facesof a polyhedron are polygons. Classifying Types of Solids Polyhedra Not Polyhedra To namea pyramid or a prism, use the shape of the base. ___________ _____________ There is 1 base of a pyramidThe two bases of a prismare congruent polygons in parallel planes. The other faces are The other faces are _____________________. __________________________. _________________ EX 1Tell whether the solid is a polyhedron. If it is, namethe polyhedron. Warning: The base is notalways the bottom!!!
A cross sectionis the intersection of a _________ and a solid.EX 2Describe the cross section formed by the intersection of the plane and the solid. What do you notice about the shape of the cross section when it is parallel to the base of the solid? EX 3Identify the three-dimensional object formed by rotating the two-dimensional object about the line. y-axis x-axis vertical line
Unit 10 –Volume of Prisms The _______________ of a solid is the number of cubic units contained in its interior. Basically, volume is the amount of three-dimensional space an object takes up. Volume DissectionTo form a prism, you can stack two-dimensional polygons on top of one another. For instance, a stack of paper becomes a rectangular prism. Each piece of paper is a cross-section of the prism. Each cross section has the same area; therefore, to find the volume you can multiply the area of a cross section by the height of the prism. Volume of a Prism The volume Vof a prism is V = ________, where Bis the _________________________________________ and h is the __________________. V = ___________ Recap: Area of a Triangle = _______________ Area of a rectangle = ______________ EX 1 Find the volume of: a)Rectangular prism b) Cube c)A right triangular prism.d) triangular prism Ex 2Write a general rule for a cube. B h h
The previous solids are called rightprisms cylinders. This is because the bases were perpendicular to the height. Solids that aren’t right are called oblique solids. How would we find their volume? Cavalieri’s Principle: The 3 solids at the right all have equal heights and equal cross-sectional area. Mathematician Bonaventura Cavalieri claimed that all three solids have the same volume. If two solids have the same height and the same cross-sectional area at every level, then they have the same __________________. c) Find the volume of the oblique prism. Unit 10 –Volume of CylindersVolume of a Cylinder The volume Vof a cylinder is V = ________ = _________________, where Bis the ______________________________, h is the __________________, and r is the ___________________. V = ___________ Exact Volume: Exact volume is when we do not _______________ our answer. This means we should leave our answer with _______. EX 1 Find the exact volume of the right cylinder.h
EX 2 a)Find the exact volume of the right cylinder. b) The volume of the right cylinder is 200cm3. Find the value of x.EX 3 The volume of a right cylinder with a radius of 6 is 1253 cm3. Find the height to the nearest hundredth.EX 4a) Find the exact volume of the oblique cylinder.
Unit 10 –Volume of Pyramids*The volume of a pyramid is _____________________ the volume of a prism with the same base area and height. Volume of a Pyramid The volume, V, of a pyramid is BhVwhere B is the ______________________________ and h is the _______________(perpendicular to the base). EX 1 Find the volume of the regular pyramids. a) b) EX 2 The Pyramid of the Sun in Teotihuacan, Mexico, is a square based pyramid with height 63m and volume 970,725m3. Find the side length of the base. EX 5 Use Cavalieri’s Principle to find the volume of the oblique solid. 6ft 8ft 10ft 3ft
Unit 10 –Volume of ConesVolume of a Cone The volume, V, of a cone is hrBhV23131where B is the ___________________________________, h is the ___________, and r is the ____________. EX 1 Find the volume of the cones. a) b) c) EX 2 The volume of a right cone is 1350m3and the radius is 18 m. Find the height. EX 3 Use Cavalieri’s Principle to find the volume of the oblique solid. EX 4You are making coffee using a cone-shaped filter. It takes 14 minutes to brew an amount of coffee that would be equal in volume to that of the filter. Find the flow rate of the coffee in cubic inches per minute.
Unit 10 –Volume of Spheres A sphereis the set of all points in space equidistant from a given ___________ called the centerof the sphere. Volume of a Sphere The volume of a sphere is 334rVwhere r is the ______________________________. EX 1 Find the exact volume of the sphere EX 2 A beach ball has a diameter of 15 in. Find the volume to the nearest hundredth. EX 3 Find the radius of a sphere with a volume of 3cm36Unit 10 –Composite SolidsComposite Solidsare solids that consist of two or more solids. Volume Addition Postulate The volume of a solid is the _________ of the volumes of all its nonoverlapping parts. EX 1 Find the exact volume of the composite solids. a) b)
EX 2 Find the volume of the pill capsule. EX 3A pool, with the dimensions shown, has been emptied for a thorough cleaning. How many gallons of water does it take to completely fill the pool with water? (Hint: 1 cubic foot = 7.4805 gallons) EX 4You want to use special filtered water to fill the pool. The water comes in 5 gallon containers. How many would you need to fill the entire pool? 4 ft 10 ft 164 ft 82 ft
Unit 10 –Modeling Objects and DensityThere are numerous objects in the world that people often need to measure in order to solve a problem. However, not all objects are perfect geometric objects; therefore, people model real-world objects with geometric objects that most closely resemble them. EX 1 Name the geometric object(s) that most closely resemble the real-world object. a. Apple b. Dumbbell c. Tent d. Grain silo e. HouseThe Densityof an object is the mass of the object per unit of _______________. The symbol used for density is the lowercase Greek letter rho: ρMathematically, Vmwhere is _______________, mis __________, and Vis _____________ Different objects or materials typically have different densities. EX 1 Calculate the density of the following objects in pounds per cm3using the provided information. Round your answers to 4 decimal places. a.1 gallon of water weighs about 8.33 lbs (Hint: 1 gallon = 3785.4118 cm3) b.The block of wood shown weighs about 164 grams (Hint: 1 pound = 453.5924 grams) c.An average American male weighs 196 lb and has a volume of .074 m3. (Hint: 1 m3= 1,000,000 cm3) 13 cm 7.45 cm 2.82 cm
Population Density is a measurement of population per unit of __________. EX 3Using the information provided, estimate the population density of the following states in people per square mile. Round your answers to the nearest hundredth. Nevada has a population of 2,758,931 people. 320 mi 483 mi 208 mi