# 2.23 Units resulting from a formula or graph

Lesson

## Determine the units from the attributes in a formula

We have learned how to make all kinds of mathematical calculations. However, if I told you the answer is $42$42, your initial response may be, "$42$42 what?" It could be $42$42 ounces or it could be $42$42 goats. That's why units of measurement are really important!

We can use units of measurement to define any physical phenomenon, such as quantities, weights, lengths, areas, volumes, and rates. Let's look at some different units of measurement now.

### Measurements of length

Length or distance is a measurement in one dimension. For example, we could measure from one point to another

Units of measurements for length include millimeters ($mm$mm), centimeters ($cm$cm), meters ($m$m), and kilometers ($km$km) or inches ($in$in), feet ($ft$ft), yards ($yd$yd), and miles ($mi$mi).

### Measurements of area

Area is a measure of two dimensions, usually called length and width, but sometimes called base and height. Area is measured in square units.

Units of measurements for area include square millimeters ($mm^2$mm2), square centimeters ($cm^2$cm2), square feet ($ft^2$ft2), and square miles($mi^2$mi2) just to name a few.

### Measurements of volume

Volume reflects the amount of space an object takes up and is a measure of three dimensions usually called length, width, and height and is measured in cubic units.

Units of measurements for volume include cubic millimeters ($mm^3$mm3), cubic inches ($in^3$in3), or cubic feet ($ft^3$ft3) to name a few.

### Measurements of mass

Units of measurement for mass include grams ($g$g), kilograms ($kg$kg), ounces ($oz$oz), pounds ($lbs$lbs) and tons ($t$t), among others. Mass is a measure of how heavy something is. We often refer to the mass of an object as its weight.

### Measurements of capacity

Capacity is similar to volume but is a measure of how much something holds, such as how much liquid will fit in a bottle.

Measures of capacity include gallons ($gal$gal), milliliters ($ml$ml), and liters ($l$l).

### Measurements in rates

A rate is a ratio between two measurements with different units. There are any number of combinations of measurement units for rates, such as miles per gallon ($mi$mi/$gal$gal), kilometers per hour ($km$km/$hr$hr) and so on.

### Units from a formula

Any formula that is made up of measurements will have units attached to each of the variables.

Here is a formula we are familiar with. The speed of an object is a measure of the distance traveled over time.

$S=\frac{d}{t}$S=dt

The units for Speed in the formula are derived from the units used for distance and time.  So if the distance is measured in kilometers and time is measured in hours, then the

$\text{Speed (unit) }=\frac{\text{distance (units) }}{\text{time (unit) }}$Speed (unit) =distance (units) time (unit)

$\text{Speed (unit) }=\frac{\text{kilometers }}{\text{hour }}$Speed (unit) =kilometers hour

$\text{Speed (unit) }=\frac{\text{kilometers }}{\text{hour }}$Speed (unit) =kilometers hour

#### Worked Examples

##### Question 1

The area of a rectangle is given by $A=l\times w$A=l×w, where $l$l is the length and $w$w is the width. Both length and width must be of the same units when performing the multiplication to find the area.

1. What would the unit for area be if the length and width are in millimeters?

m2

A

cm-km

B

mm2

C

mm

D

m2

A

cm-km

B

mm2

C

mm

D

##### Question 2

Adam plotted a point to represent material purchases ($x$x) and the costs involved ($y$y). When Adam bought $140$140cm of material, it cost $\$2.202.20.

1. What unit is the $x$x-axis using?

cost in dollars of material purchased per meter

A

cost in dollars of material purchased per sale

B

meters of material purchased

C

centimeters of material purchased

D

cost in dollars of material purchased per meter

A

cost in dollars of material purchased per sale

B

meters of material purchased

C

centimeters of material purchased

D
2. What unit is the $y$y-axis using?

cost in dollars of material purchased

A

centimeters of material purchased

B

cost in cents of material purchased

C

meters of material purchased

D

cost in dollars of material purchased

A

centimeters of material purchased

B

cost in cents of material purchased

C

meters of material purchased

D

## Dimensional analysis (construction of units)

Remember!

Length has $1$1 dimension, therefore the power for the units is $1$1 such as $in$in$m$m, or $ft$ft.

Area has $2$2 dimensions, therefore the power for the units is $2$2 such as $in^2$in2$m^2$m2, or $ft^2$ft2.

Volume has $3$3 dimensions, therefore the power for the units is $3$3 such as $in^3$in3$m^3$m3, or $ft^3$ft3.

#### Worked examples

##### question 3

If $F=3q+r$F=3q+r, where both $q$q and $r$rrepresent lengths, what does $F$F represent?

Think: Consider each part of this equation. The power of each part is just $1$1 at the moment. Does the addition of $3q$3q and $r$r change the power?

Do: As you are adding, there is no change to the power. Therefore this is still a measurement of length.

##### question 4

If $F=\frac{s^3}{p}$F=s3p where both s and p represent lengths, what does $F$F represent?

Think: $s$s has a power of $3$3, so it must be a volume. However, it is then divided by a length ($F$F). What happens to the power in this case?

Do: If a volume (power of $3$3) is divided by a length (power of $1$1), then the resulting power is $2$2 which represents an area.

#### Practice questions

##### question 5

If $\text{work}=\text{force}\times\text{distance}$work=force×distance, what are the dimensions of work?

1. $\text{time}\times\text{length}^2\times\text{mass}^2$time×length2×mass2

A

$\frac{\text{mass}^2\times\text{time}}{\text{length}^2}$mass2×timelength2

B

$\frac{\text{mass}\times\text{length}}{\text{time}}$mass×lengthtime

C

$\frac{\text{mass}\times\text{length}^2}{\text{time}^2}$mass×length2time2

D

$\text{time}\times\text{length}^2\times\text{mass}^2$time×length2×mass2

A

$\frac{\text{mass}^2\times\text{time}}{\text{length}^2}$mass2×timelength2

B

$\frac{\text{mass}\times\text{length}}{\text{time}}$mass×lengthtime

C

$\frac{\text{mass}\times\text{length}^2}{\text{time}^2}$mass×length2time2

D

##### Question 6

This graph's slope has a unit of km/h.

1. What is the unit on the $x$x-axis?

km/h

A

hours

B

kilometers

C

km/h

A

hours

B

kilometers

C
2. What is the unit on the $y$y-axis?

hours

A

km/h

B

kilometers

C

hours

A

km/h

B

kilometers

C

### Outcomes

#### N.Q.A.2

Define appropriate quantities for the purpose of descriptive modeling.