Solid Rotations
This 3D shape is formed when a rectangle is rotated around one of its longer sides. (along the axis of rotation)
What is a cylinder?
This shape is the cross section of a sphere cut horizontally.
What is a circle?
If a shape is scaled by a factor of 3, this is how much larger its area is.
What is 9 times larger?
What happens to a solid’s volume when scaled by a factor of 2?
Hint: If K=2, what is the factor for VOLUME?
What is the volume increases by 8 (2³)
K=3; K3=27
Define density in terms of volume.
(What is the formula for density?)
Density = mass ÷ volume.
This solid is formed when you rotate a right triangle around one of its legs.
What is a (right) cone?
A vertical slice through the center of a cylinder creates this shape.
What is a rectangle?
This kind of graph can represent the relationship between scale factors and area.
Hint: If the factor for area is K2, what operation do we apply to find K? --- "We take the _____ ____!"
What is a square root graph?
This is the formula for the volume of any prism or cylinder.
2x DOUBLE JEOPARDY 2x
What is V = Bh?
400 POINTS!!!
This is the density of a block that weighs 100g and occupies 25cm³.
What is 4 g/cm³?
This solid is formed when a semicircle is rotated around its diameter.
2X DOUBLE JEOPARDY 2X
What is a sphere?
600 POINTS!!!
This is the shape you get if you cut a cone horizontally.
A circle (that is smaller than the base).
A triangle has an area of 10 in². If it’s scaled by 0.5, this is the new area.
What is 2.5 in²?
If two solids have the same cross-sectional area at the same height, this principle says they have the same volume.
What is Cavalieri’s Principle?
This is the formula for the volume of any pyramid.
What is V = ⅓Bh?
Let's call back to Exit Ticket #1 and the man with the telescope. This 2D shape produced the multiple cylinders forming the telescope.
What is a rectangle?
This shape results from slicing a cyclinder diagonally.
What is an oval?
A square with side length 4ft is dilated by a factor of √2. This is the area of the new square.
2x DOUBLE JEOPARDY 2X
What is 32 ft2?
800 POINTS!!!
This is the volume of a cone with radius 3 and height 6.
What is 18π?
If a cube's side length is tripled (side length * 3), by this factor does its volume increase.
Hint: If K=3, what is the factor for VOLUME?
What is 27 or (3³)?
27 = 3*3*3
K=3; K3=27
A construction worker uses a drill bit shaped like a triangle. When the bit spins, it carves out a hole in the shape of a 3D solid. This is the shape of the hole created.
Hint: What does the original triangle represent?
What is a cone?
The triangle represents the cross-section.
This 3D solid has all circular cross sections, regardless of the slicing plane’s angle.
Explain how scale factors affect the area of a pyramid’s cross sections.
!!FINAL JEOPARDY!!
Cross sections are scaled versions of the base, and the area is multiplied by the square of the scale factor (k²).
A solid is scaled by k. This is how the surface area and volume change.
What is the surface area scales by k², volume by k³?
You have a solid metal object with a mass of 300 grams and a volume of 100 cm³. Will it float or sink in water? Justify your answer using density.
Water's Density = 1g/cm3
First, calculate the object’s density:
Density = Mass ÷ Volume = 300 g ÷ 100 cm³ = 3 g/cm³.
Water has a density of 1 g/cm³, and since 3 > 1, the object will sink in water because it’s denser than water.