Problem 1
Answers:
- Cone: Triangle
- Cylinder: Rectangle
- Sphere: Circle
------------------------
Explanation:
Imagine a fan blade rotating. As it rotates really quickly, it occupies a 3D space. Or perhaps imagining rotating eggbeaters is a better thing to have in mind. Those blades spin really fast to form a 3D shape. So that's what's going on with this problem. If you spin a triangle around an axis, a cone will form. The base of the triangle forms the base of the cone. Half the base of the triangle is the radius of the cone. The height of the triangle is the height of the cone.
Similarly, a rectangle rotated around its center axis forms a cylinder. Think of a revolving door. That's often a classic example. Finally, a circle spun around any of its diameters will form a sphere.
========================================================
Problem 2
Answers:
- Object 1: Cup = Cylinder
- Object 2: Ball = Sphere
- Object 3: Funnel = Cone
- Object 4: Paper towel roll = Cylinder
-------------------------------
Explanation:
Pick objects around your house that are fairly simple in design and resemble either a cylinder, sphere, or cone. A cup resembles a cylinder just it doesn't have a top to it. A ball is a 3D sphere that has an interior to it, unless the ball is filled with air. A funnel resembles a cone due to its triangular shape of sorts. A paper towel roll (the card board piece or the cardboard plus the actual paper towels) resemble a cylinder as well. You can pick any four other objects you have around your house such as a cylindrical speaker, batteries (shaped as a cylinder), soup cans (also cylinders), and so on.
========================================================
Problem 3
Answers:
- a) volume of cylinder = pi*r^2*h
- b) volume of sphere = (4/3)*pi*r^3
- c) volume of pyramid = (1/3)*L*W*H
- d) volume of cone = (1/3)*pi*r^2*h
-------------------------------
Explanation:
These formulas are what you should memorize or have handy on a flashcard. It is possible to derive them, but it would require calculus to do so. Also, such a process is quite length. So memorization or flashcards are the better way to go.
For the cylinder and cone, r is the radius and h is the height. Note how the formulas are nearly identical except the cone has the (1/3) term. This shows that the cone's volume is 1/3 that of the cylinder's volume where they both have the same radius r and height h.
The sphere also deals with the radius r, but we don't have to worry about the height of the sphere (it's really just 2r).
For the pyramid, L,W,H represent the length width and height respectively. The length and width form the base of the pyramid. Recall that L*W*H is the volume of a rectangular block. The volume of a pyramid is 1/3 of that rectangular block's volume.
