Lesson 17Scaling One Dimension
Learning Goal
Let’s see how changing one dimension changes the volume of a shape.
Learning Targets
I can create a graph the relationship between volume and height for all cylinders (or cones) with a fixed radius.
I can explain in my own words why changing the height by a scale factor changes the volume by the same scale factor.
Warm Up: Driving the Distance
Problem 1
Here is a graph of the amount of gas burned during a trip by a tractor-trailer truck as it drives at a constant speed down a highway:
At the end of the trip, how far did the truck drive, and how much gas did it use?
If a truck traveled half this distance at the same rate, how much gas would it use?
If a truck traveled double this distance at the same rate, how much gas would it use?
Complete the sentence:
is a function of
Activity 1: Double the Edge
Problem 1
There are many right rectangular prisms with one side of length 5 units and another side of length 3 units. Let
Write an equation that represents the relationship between
and . Graph this equation and label the axes.
What happens to the volume if you double the side length
? Where do you see this in the graph? Where do you see it algebraically?
Print Version
There are many right rectangular prisms with one edge of length 5 units and another edge of length 3 units. Let
Write an equation that represents the relationship between
and . Graph this equation and label the axes.
What happens to the volume if you double the edge length
? Where do you see this in the graph? Where do you see it algebraically?
Activity 2: Halve the Height
Problem 1
There are many cylinders with radius 5 units. Let
Write an equation that represents the relationship between
and . Use 3.14 as an approximation of . Graph this equation and label the axes.
What happens to the volume if you halve the height,
? Where can you see this in the graph? How can you see it algebraically?
Print Version
There are many cylinders with radius 5 units. Let
Write an equation that represents the relationship between
and . Use 3.14 as an approximation of . Graph this equation and label the axes.
What happens to the volume if you halve the height,
? Where can you see this in the graph? How can you see it algebraically?
Are you ready for more?
Problem 1
Suppose we have a rectangular prism with dimensions 2 units by 3 units by 6 units, and we would like to make a rectangular prism of volume 216 cubic units by stretching one of the three dimensions.
What are the three ways of doing this? Of these, which gives the prism with the smallest surface area?
Repeat this process for a starting rectangular prism with dimensions 2 units by 6 units by 6 units.
Can you give some general tips to someone who wants to make a box with a certain volume, but wants to save cost on material by having as small a surface area as possible?
Activity 3: Figuring Out Cone Dimensions
Problem 1
Here is a graph of the relationship between the height and the volume of some cones that all have the same radius:
What do the coordinates of the labeled point represent?
What is the volume of the cone with height 5? With height 30?
Use the labeled point to find the radius of these cones. Use 3.14 as an approximation for
. Write an equation that relates the volume
and height .
Lesson Summary
Imagine a cylinder with a radius of 5 cm that is being filled with water. As the height of the water increases, the volume of water increases.
We say that the volume of the water in the cylinder,
This equation represents a proportional relationship between the height and the volume. We can use this equation to understand how the volume changes when the height is tripled.
The new volume would be
Remember that proportional relationships are examples of linear relationships, which can also be thought of as functions. So in this example