When we measure something with a ruler, we are making a one dimensional measurement as there is one direction of our measurement (left and right).
When we measure the square feet of a room, we are making a two dimensional measurement, as the measurement has two directions (left and right, up and down).
Volume is a measurement of three dimensions. Volume measures space that has three directions (left and right, up and down, forward and backward).
1.
https://www.youtube.com/watch?v=jl5uHvfdXeA
A plumber has been asked to choose and install a water heater for
a family. The dimensions of the water heater will be constrained based
on the location of the new water heater in the home. The cylindrical
tank that fits this location has a height of 61 inches and a diameter of
20 inches. What is the volume of the water tank?
The volume of an object can be found by finding the area of the base of the object and multiplying by the height. The base of a cylinder is a circle. The area of this circle is \(\pi (10)^2 \text{ in.}^2 \approx 314.16 \text{ in.}^2\). The height is 61 inches. The total volume is therefore
\[314.16 \text{ in.}^2 \cdot 61 \text{ in.} = 19,163.76 \text{ in.}^3\]
Note the dimensions of the units. If we multiply \(x \cdot x^3\), the result is \(x^{1 + 3} = x^4\). Similarly, when we multiply \(\text{ in.}^2\) and \(\text{ in.}\), we get \(\text{ in.}^3\).
Water tank capacity is usually measured in gallons, though, and not cubic inches. To convert from cubic inches to gallons, we will use the fact that 1 gallon is 231 cubic inches. We will multiply our result by \(\frac{1 \text{ gallon}}{231 \text{ in.}^3}\) so that the inches units cross cancel.
\[19,163.76 \text{ in.}^3 \cdot \frac{1 \text{ gallon}}{231 \text{ in.}^3} =\]
\[19,163.76 \cancel{\text{ in.}^3} \cdot \frac{1 \text{ gallon}}{231 \cancel{\text{ in.}^3}} =\]
\[\frac{19,163.76}{231} \text{ gallons} =\]
\[82.96 \text{ gallons}\]
2.
https://youtu.be/jl5uHvfdXeA?si=B8yiRB8GHfI7MULh&t=216
A family is trying to select a small swimming pool. They are
interested in the volume of a swimming pool. For some reason, though,
the website is stating the dimensions in two different units. The
dimensions listed on the website are that the pool is 160 cm in diameter
and 12 inches in height. What is the volume of the pool in gallons?
We can convert the centimeters dimension to inches. 1 inch is 2.54 cm.
\[160\text{ cm} \cdot \frac{1 \text{ in.}}{2.54 \text{ cm}} =\]
\[160\cancel{\text{ cm}} \cdot \frac{1 \text{ in.}}{2.54 \cancel{\text{ cm}}} =\]
\[\frac{160}{2.54} \text{ inches} \approx\]
\[63 \text{ inches}\]
A pool has a cylinder shape. We can now find the volume by finding the area of the base and multiplying by the height. The diameter is 63 inches, so the radius is 31.5 inches.
\[\pi \cdot (31.5)^2 \cdot 12 \text{ in.}^3 =\]
\[\pi \cdot 992.25 \cdot 12 \text{ in.}^3 \approx\]
\[37,406.94 \text{ in.}^3\]
1 gallon is 231 cubic inches.
\[37,406.94 \text{ in.}^3 \cdot \frac{1 \text{ gallon}}{231 \text{ in.}^3} =\]
\[37,406.94 \cancel{\text{ in.}^3} \cdot \frac{1 \text{ gallon}}{231 \cancel{\text{ in.}^3}} =\]
\[\frac{ 37,406.94}{231} \text{ gallons}\approx\]
\[161.93 \text{ gallons}\]
3.
https://youtu.be/jl5uHvfdXeA?si=pgI8me-k49dkD6hH&t=385
A punching bag is roughly cylindrical. The diameter of the base is
14 inches and the height is 42 inches. One bag of punching bag filler
will fill 14 liters. How many bags of filler do we need?
First, we can calculate the volume of the cylinder.
\[\pi \cdot 7^2 \cdot 42 \text{ in}^3 =\]
\[\pi \cdot 49 \cdot 42 \text{ in}^3 \approx\]
\[6,465.3 \text{ in}^3\]
The bags of filler are in liters. 1 liter is about 61.02 cubic inches.
\[6,465.3 \text{ in}^3 \cdot \frac{1 \text{ liter}}{61.02 \text{ in.}}^3 =\]
\[6,465.3 \cancel{\text{ in}^3} \cdot \frac{1 \text{ liter}}{61.02 \cancel{\text{ in.}}^3} =\]
\[\frac{6,465.3}{61.02}\text{ liters} =\]
\[105.95\text{ liters}\]
One bag fills 14 liters, so we’ll divide the number of liters we need to fill by the number of liters per bag.
\[105.95\text{ liters} \div 14 \text{ liters per bag} \approx\]
\[7.6 \text{ bags}\]
This person should purchase 8 bags of filler to make sure they have enough.
1. A plumber has been asked to choose and install a water heater for a family. The dimensions of the water heater will be constrained based on the location of the new water heater in the home. The cylindrical tank that fits this location has a height of 58 inches and a diameter of 18 inches. What is the volume of the water tank?
2. A family is trying to select a small swimming pool. They are interested in the volume of a swimming pool. For some reason, though, the website is stating the dimensions in two different units. The dimensions listed on the website are that the pool is 200 cm in diameter and 10 inches in height. What is the volume of the pool in gallons?
3. A punching bag is roughly cylindrical. The diameter of the base is 12 inches and the height is 48 inches. One bag of punching bag filler will fill 12 liters. How many bags of filler do we need?
1. What is the relationship between the height of a cylinder and the volume of a cylinder? Is it linear, exponential, etc.?
2. Do you realize that we’re floating in space?