1.7: Formulas

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You may use a calculator throughout this module if needed.

A baby drinking formula from a bottle — Photo by Rainier Ridao on Unsplash.

A formula is an equation or set of calculations that takes a number (or numbers) as input, and produces an output. The output is often a number, but it could also be a decision such as yes or no.

Each unknown number in a formula is called a variable because its value can vary. A variable is usually represented with a letter of the alphabet. To evaluate a formula, we substitute a number (or numbers) into the formula and then perform the steps using the order of operations.

Formulas with One Input

Many formulas will have just one input variable. Note: When a number is written directly next to a variable, it indicates multiplication. For example, $0.24w$ means $0.24\cdot{w}$.

Exercises

The formula $C=0.24w+1.26$ gives the cost, in dollars, of mailing a large envelope weighing $w$ ounces through the USPS.^[1]

Find the cost of mailing a $6$-ounce envelope.
Find the cost of mailing a $12$-ounce envelope.

Radio Cab charges the following rates for a taxi ride: a fixed fee of $$3.80$ to get in the taxi, plus a rate of $$2.80$ per mile.^[2] The total cost, in dollars, of a ride $m$ miles long can be represented by the formula $C=3.80+2.80m$.

Find the cost of a $5$-mile ride.
Find the cost of a $7.5$-mile ride.
Find the cost of getting in the taxi, then changing your mind and getting out without riding anywhere.

The number of members a state has in the U.S. House of Representatives can be approximated by the formula $R=P\div{0.76}$, where $P$ is the population in millions.^[3] The 2020 populations of three states are as follows:^[4]

Oregon	$4.2\text{ million}$
Washington	$7.7\text{ million}$
California	$39.6\text{ million}$

Round all answers to the nearest whole number.

How many U.S. Representatives does Oregon have?
How many U.S. Representatives does Washington have?
How many U.S. Representatives does California have?

The number of electoral votes a state has can be approximated by the formula $E=P\div{0.76}+2$, where $P$ is the population in millions.

How many electoral votes does Oregon have?
How many electoral votes does Washington have?
How many electoral votes does California have?

Temperature Conversions

You probably know that $32$° Fahrenheit is the freezing point of water, and $212$° Fahrenheit is the boiling point of water. The Celsius equivalents are $0$°C and $100$°C. The formulas shown below allow us to convert between the two temperature scales.

Temperature Formulas

$F=\dfrac{9}{5}C+32$

$F=1.8C+32$

$C=\dfrac{5}{9}(F-32)$

$C=(F-32)\div{1.8}$

Exercises

The record high temperature in Portland, Oregon occurred during the “heat dome” event in June, 2021. As shown, it was $46$°C at your author’s house and the squirrels were laying low. Convert this temperature to Fahrenheit.

Based on the evidence, the melting point of squirrel is right around 46°C.

The traditionally accepted “normal” body temperature of a human^[5] is $98.6$°F. What is this temperature in Celsius?
The FDA recommends that a freezer be set below $–18$°C. What is the Fahrenheit equivalent?
A package of frozen pancakes from IKEA calls for the oven to be set to $392$°F. IKEA is based in Sweden, and this temperature clearly was originally calculated in Celsius. What is the corresponding Celsius temperature?

$IKEA-pancakes-Celsius-cropped-300x54.jpg$

Formulas with More than One Input

Some formulas have more than one input variable. Just pay attention to which number goes in for each variable.

Exercises

When a patient’s blood pressure is checked, they are usually told two numbers: the systolic blood pressure (SBP) and the diastolic blood pressure (DBP). The mean arterial pressure (MAP) can be estimated by the following formula: $MAP=\dfrac{SBP+2\cdot{DBP}}{3}$. (The units are mm Hg.) Calculate the mean arterial pressure for each patient.

SBP = $120$, DBP = $75$
SBP = $140$, DBP = $90$

$a three-dimensional rectangular box with the bottom front edge labeled l, the bottom right edge labeled w, and the vertical right edge labeled h$ UPS uses the following formula^[6] to determine the “measurement” of a package with length $l$, width $w$, and height $h$: $m=l+2w+2h$. Determine the measurement of a package with the following dimensions.

length $18$ inches, width $12$ inches, height $14$ inches
length $16$ inches, width $14$ inches, height $15$ inches

Formulas with a Yes or No Answer

Some formulas give a yes or no answer: success or failure, approved or disapproved, etc. After calculating the result from the formula, we need to compare it to a given number to see whether the result is within a specified range.

Exercises

$Eggs.png$ In Australia, a chicken egg is designated “large” if its mass, in grams, satisfies the following formula: $|m-54.1|\leq4.1$. Determine whether each egg qualifies as large.^[7]

Egg 1’s mass is $57.8$ grams.
Egg 2’s mass is $58.3$ grams.
Egg 3’s mass is $49.8$ grams.
Egg 4’s mass is $50.0$ grams.

Source: https://pe.usps.com/text/dmm300/Notice123.htm#_c037 ↵
Source: https://www.radiocab.net/services-radio-cab/↵
The value 0.76 comes from dividing the total U.S. population in 2020, around 331 million people, by the 435 seats in the House of Representatives. ↵
Source: https://www.census.gov/data/tables/2020/dec/2020-apportionment-data.html ↵
See this article for a counterargument: www.nytimes.com/2023/10/12/well/live/fever-normal-body-temperature.html↵
Source: www.ups.com/us/en/help-center/packaging-and-supplies/prepare-overize.page↵
Source: https://en.Wikipedia.org/wiki/Chicken_egg_sizes ↵

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Oregon	\(4.2\text{ million}\)
Washington	\(7.7\text{ million}\)
California	\(39.6\text{ million}\)