1.4: Dimensional Analysis

Rates

In the last section we introduced the definition of a rate. Remember a rate is a comparison of two quantities with different units, such as miles per gallon, words per minute, or dollars per ounce. In many applications we may not need to go to the step of writing a proportion equation if we are given a rate.

When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator. For example if Darla drove her car 525 miles in 10 hours we can express this information as a rate $\frac{\text{525 miles}}{\text{10 hours}}$ or as $\frac{\text{52.5 miles}}{\text{1 hour}}$ as a unit rate.

The advantage of a unit rate is that if we multiply that unit rate by the amount of time (in hours) we will get the distance traveled in that time. This removes a step in setting up a proportion equation to solve. This strategy of multiplying the rate by the known quantity can be used as long as we make sure to have the unit rate denominator be the same units as what we are multiplying by (as we will see in the next example).

Many problems involving rates can be solved by using dimensional analysis, the process of multiplying a quantity by rate(s) to change the units. For example if we know that a car is traveling at a constant rate of 60 miles per hour and travels for 3 hours we can find the distance traveled by multipling the rate by the time to find that the car traveled 180 miles. This process leads us to the general formula for distance traveled (D) over time (T) with a constant rate (R) as $D=R\cdot T$. One caution with using the formula is that the rate and time must be using the same units of time. We will see how to handle the situation when they are not the same units in a later example.

Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.

Convert Between Common Measurements

You will often encounter situations where converting between different units of measurement is necessary. This is where dimensional analysis becomes a powerful tool. While most countries around the world use the Metric system, the United States primarily uses a different system known as the United States customary units. This system is sometimes referred to as English Units or the Imperial System, although it’s worth noting that even England now uses the metric system.

You might be surprised to learn that the metric system is already a significant part of daily life in the United States. Even though we still buy milk by the gallon and measure house sizes in square feet, metric units are more common than you might think. Here are some examples:

• Soda is often sold in 2-liter bottles
• Medicine dosages are typically measured in milligrams or milliliters
• Athletes run 100-meter dashes in track events
• Car engine sizes are usually given in liters
• Ammunition is often described using metric calibers (e.g., 9mm)
• Scientific measurements almost always use metric units

Understanding how to convert between these two systems is crucial, whether you’re traveling abroad, pursuing a career in science or engineering, or simply trying to understand product specifications. In the following sections, we’ll explore various conversion techniques and practice problems to help you master this essential skill.

In the video, Neil DeGrasse Tyson and comedian co-host Chuck Nice provide an amusing perspective on the metric system. Neil deGrasse Tyson Explains the Metric System

Here are some unit conversions for reference that has both the Metric system and US system. If you plan on traveling outside of the US you will most likely see measurements given in the metric system.

In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions. We’ll restate the Identity Property of Multiplication here for easy reference.

To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that 1 foot is equal to 12 inches, so we can write 1 as the fraction $\frac{1\text{ ft}}{12\text{ in}}$. When we multiply by this fraction, we do not change the value but just change the units.

But $\frac{12\text{ in}}{1\text{ ft}}$ also equals 1. How do we decide whether to multiply by $\frac{1\text{ ft}}{12\text{ in}}$ or $\frac{12\text{ in}}{1\text{ ft}}$? We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert 60 inches to feet. If we choose the fraction that has the inches in the denominator, we can then eliminate the inches.

$$\text{60 }\bcancel{in}\cdot \frac{1\text{ ft}}{12\bcancel{in}}=5\text{ ft}$$

On the other hand, if we wanted to convert 5 feet to inches, we would choose the fraction that has feet in the
denominator.

$$5\bcancel{\text{ft}}\cdot \frac{12\text{ in}}{1\bcancel{\text{ft}}}=60\text{ in}$$

We treat the unit words like factors and ‘divide out’ common units like we do common factors.

In some situations we are dealing with units that are squared or cubed if the application is involving area or volume. In those cases we treat an area unit like ft² as the units of feet times itself or feet*feet and will need a conversion for each one in order to find the new units we are looking for.

Lets take a look at an application that has mixed units now.

Further Applications With Unit Conversions

Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean.

Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor.

Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter.

It may be helpful to recall some formulas for areas and perimeter of a few basic two dimensional shapes.

The rectangular solid shown in the Figure below has length 4 units, width 2 units, and height 3 units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it
contains. This 4 by 2 by 3 rectangular solid has 24 cubic units.

Altogether there are 24 cubic units. Notice that 24 is the length×widthxheight.

$$\begin{array}{l}V=L\cdot W\cdot H\\ 24=4\cdot 2\cdot 3\end{array}$$

The volume, V, of any rectangular solid is the product of the length, width, and height.

$$V=L\cdot W\cdot H$$

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, B, is equal to length×width.

$$B=L\cdot W$$

We can substitute B for L·W in the volume formula to get another form of the volume formula.

$$\begin{array}{l}V=L\cdot W\cdot H\\ V=\left(L\cdot W\right)\cdot H\\ V=B\cdot H\end{array}$$

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the 4×2×3 rectangular solid we started with.

Volume of the rectangular solid is the area of the base multiplied by the height.

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

$$\begin{array}{ccccccc}{A}_{\text{front}}=L\times W\hfill & & & A_{\text{side}}=L\times W\hfill & & & A_{\text{top}}=L\times W\hfill \\ A_{\text{front}}=4·3\hfill & & & A_{\text{side}}=2·3\hfill & & & A_{\text{top}}=4·2\hfill \\ A_{\text{front}}=12\hfill & & & A_{\text{side}}=6\hfill & & & A_{\text{top}}=8\hfill \end{array}$$

Notice for each of the three faces you see, there is an identical opposite face that does not show.

$$\begin{array}{l}S=(\text{front}+\text{back})\text{+}(\text{left side}+\text{right side})+(\text{top}+\text{bottom})\\ S=(2·\text{front})+(\text{2}·\text{left side})+(\text{2}·\text{top})\\ S=2·12+2·6+2·8\\ S=24+12+16\\ S=52\text{sq. units}\end{array}$$

The surface area S of the rectangular solid shown in Figure above is 52 square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see Figure below). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

For each face of the rectangular solid facing you, there is another face on the opposite side. There are 6 faces in all.

Three Dimensional

Rectangular Box	Cylinder
$$\begin{array}{l}\text{Volume: }V=L\cdot W\cdot H\\ \text{Surface Area: }S=2\mathrm{LW}+2\mathrm{HW}+2\mathrm{HL}\end{array}$$	$\begin{array}{l}\text{Volume: }V=\pi r^{2}H\\ \text{Surface Area: }S=2\pi r^2+2\pi rH\end{array}$

We don’t always need to have an exact measurements if there is a way to create our own units to scale items as shown in the next example below.

Example 9

A company makes regular and jumbo marshmallows.   The
regular marshmallow has 25 calories.  How many calories will the jumbo marshmallow have?

---

Solution

We would expect the calories to scale with volume.  Since the marshmallows have cylindrical shapes, we can use that formula to find the volume.  From the grid in the image, we can estimate the radius and height of each marshmallow.

The regular marshmallow appears to have a diameter of about 3.5 units, giving a radius of 1.75 units, and a height of about 3.5 units.  The volume is about

$$\text{π}{\left(1.75\right)}^2\left(3.5\right)=33.7{\text{ units}}^3$$.

The jumbo marshmallow appears to have a diameter of about 5.5 units, giving a radius of 2.75 units, and a height of about 5 units.  The volume is about

$$\text{π}{\left(2.75\right)}^2\left(5\right)=118.8{\text{ units}}^3$$.

We could now set up a proportion, or use rates.  The regular marshmallow has 25 calories for 33.7 cubic units of volume.  The jumbo marshmallow will have:

$$118.8{\text{ units}}^3\cdot \frac{25\text{ calories}}{33.7{\text{ units}}^3}=88.1\text{ calories}$$

It is interesting to note that while the diameter and height are about 1.5 times larger for the jumbo marshmallow, the volume and calories are about 1.5³ = 3.375 times larger.

Video Solution (2 mins and 19 seconds – CC)

Exercises

1. Find a unit rate: You drove 360 miles on 20 gallons of fuel.
2. Find a unit rate: You bought 10 pounds of potatoes for $4.
   
   Answer (click to Show/Hide)

   $4/10 pounds = $0.40 per pound   (or 10 pounds/$4 = 2.5 pounds per dollar)

3. Find a unit rate: Joel ran 1500 meters in 4 minutes, 45 seconds.
4. Convert 8 feet to inches.
5. Convert 2 miles to yards.
6. Convert 40 miles to kilometers.
   
   Answer (click to Show/Hide)

   Using just the conversion formulas found in the table we have:

   $$40\text{ miles }\cdot \frac{\text{5280 ft}}{1\text{ mile}}\cdot \frac{12\text{ in}}{1\text{ ft}}\cdot \frac{2.54\text{ cm}}{1\text{ in}}\cdot \frac{\text{1 m}}{100\text{ cm}}\cdot \frac{1\text{ km}}{1000\text{ m}}=64.38\text{ km}$$

7. Convert 6 kilograms to grams.
8. Convert 2.5 gallons to ounces.
9. An outdoor track is 400 m long. If you run 10 laps around the track, how many kilometers have you run?
10. An aspirin tablet is 650 mg. If you take 2 aspirin twice in one day, how many grams of aspirin have you taken?
11. A wire costs $2 per meter.  How much will 3 kilometers of wire cost?
12. Sugar contains 15 calories per teaspoon.  How many calories are in 1 cup of sugar?
13. A car is driving at 100 kilometers per hour.  How far does it travel in 2 seconds (in meters)?
14. A chain weighs 10 pounds per foot.  How many ounces will 4 inches weigh?
    
    Answer (click to Show/Hide)

    First set up a proportional equation.

    $$\begin{array}{l}\frac{10lbs}{1ft}=\frac{xounces}{4in.}\\ xoz=\frac{10lbs}{1ft}*4in,\text{ Now we have to worry about the units}\\ xoz=\frac{10lbs*\frac{16oz}{1lb}}{1ft*\frac{12in}{1ft}}*4in\\ xoz=\frac{160oz}{12in}*4in\\ xoz=\frac{160oz}{3}\approx 53.33oz\end{array}$$

15. A recipe for sabayon calls for 2 egg yolks, 3 tablespoons of sugar, and ¼ cup of white wine.
    After cracking the eggs, you start measuring the sugar, but accidentally put in 4 tablespoons of
    sugar.  How can you compensate?
16. The store is selling lemons at 2 for $1. Each yields about 2 tablespoons of juice.  How much will it
    cost to buy enough lemons to make a 9-inch lemon pie requiring ½ cup of lemon juice?
17. A car is driving at 45 miles per hour. How far does it travel in 2 seconds?
18. Convert 245 in² to ft². Round your answer if needed to nearest tenth place.
19. Convert 12,000 m² to km². Round your answer if needed to nearest tenth place.
20. Convert 3 yd³ to ft³. Round your answer if needed to nearest tenth place.
21. Find the perimeter and area of a rectangle with width 2 feet and length 35 inches.
22. Find the length of a rectangle with a perimeter of 80 inches and width of 25 inches.
    
    Answer (click to Show/Hide)

    L=15 in

23. The area of a rectangular region is 221 ft². If the length is 156 inches, then what is the perimeter?
24. The area of a rectangular room is 168 square feet. The length is 4 yds. What is the width in yds?
25. The area of a triangular window is 90 square meters. The base of the window is 15 yards. What is the window’s height in yards?
26. Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 50 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 10 feet. How long can he make the other side if he wants to use the entire roll of fence?
27. A childs sandbox needs to be filled with sand. The dimensions of the box are 4 ft by 4 ft and is 16 inches deep. Find the volume rounded to the nearest cubic ft.
28. A box is created from a 6 ft by 4 ft piece of cardboard. To create the box a square piece is cut out from each corner. The square piece measures 2 in by 2 in. The sides are then folded up and taped together. Find the Volume of the box rounded to the nearest tenth of a cubic ft.
29. A circular pool needs the to be repainted inside. The diameter of the pool is measured to be 14 ft and the depth is 5 ft and 8 inches. Find the surface area for the interior of the pool rounded to the nearest cubic ft (keep in mind the sides and bottom is the only region needing to be painted).
30. A local pool used by lap swimmers has dimensions 25 yd by 20 yd and is 5 feet deep. Find the cost for filling the pool if the city charges $1.50 per 1000 gallons. Use the conversion 1 gallon = 0.134 cubic ft.

Attributions

• This page contains modified content from David Lippman, “Math In Society, 2nd Edition.” Licensed under CC BY-SA 4.0.
• This page contains modified content and images from “OpenStax PreAlgebra” by Lynn Marecek, MaryAnne Anthony-Smith. Licensed under CC BY-NC 4.0 / A derivative from the original work
• This page contains modified content and images from “Contemporary Mathematics” by Donna Kirk and is licensed under CC BY-NC 4.0 / A derivative from the original work.
• This page contains content by Robert Foth, Math Faculty, Pima Community College, 2021. Licensed under CC BY 4.0.