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Illustrative Math
Grade 8
Let’s see how changing one dimension changes the volume of a shape.
Illustrative Math Unit 8.5, Lesson 18 (printable worksheets)
There are many rectangular prisms that have a length of 4 units and width of 5 units but differing heights. If h represents the height, then the volume V of such a prism is
V = 20h
The equation shows us that the volume of a prism with a base area of 20 square units is a linear function of the height. Because this is a proportional relationship, if the height gets multiplied by a factor of a, then the volume is also multiplied by a factor of a:
V = 20(ah)
What happens if we scale two dimensions of a prism by a factor of a? In this case, the volume gets multiplied by a factor of a twice, or a^{2}.
For example, think about a prism with a length of 4 units, width of 5 units, and height of 6 units. Its volume is 120 cubic units since 4 • 5 • 6 = 120. Now imagine the length and width each get scaled by a factor of a, meaning the new prism has a length of 4a, width of 5a, and a height of 6. The new volume is 120a^{2} cubic units since 4a · 5a · 6.
A similar relationship holds for cylinders. Think of a cylinder with a height of 6 and a radius of 5.
The volume would be 150π cubic units since π · 5^{2} · 6 = 120π. Now, imagine the radius is scaled by a factor of a. Then the new volume is π · (5a)^{2} · 6 = π 25a^{2} · 6 or 150a^{2}π cubic units. So scaling the radius by a factor of a has the effect of multiplying the volume by a^{2}!
Why does the volume multiply by a^{2} when only the radius changes? This makes sense if we imagine how scaling the radius changes the base area of the cylinder. As the radius increases, the base area gets larger in two dimensions (the circle gets wider and also taller), while the third dimension of the cylinder, height, stays the same.
m, n, a, b, and c all represent positive integers. Consider these two equations: m = a + b + c n = abc
Clare sketches a rectangular prism with a height of 11 and a square base and labels the edges of the base s. She asks Han what he thinks will happen to the volume of the rectangular prism if she triples s.
Han says the volume will be 9 times bigger. Is he right? Explain or show your reasoning.
A cylinder can be constructed from a piece of paper by curling it so that you can glue together two opposite edges (the dashed edges in the figure).
Double x
It does not matter.
To increase the volume, we would double y instead of double x.
There are many cones with a height of 7 units. Let r represent the radius and V represent the volume of these cones.
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