1.10 Formulas

section 1.10 Learning Objectives

1.10: Formulas

  • Evaluate a formula for given values
  • Solve for a specified variable in a formula

 

Consider the following problem in the video:

We could have isolated the w in the formula for perimeter before we solved the equation, and if we were going to use the formula many times, it could save a lot of time. The next example shows how to isolate a variable in a formula before substituting known dimensions or values into the formula.

Example 1

Solve for the variable, [latex]w[/latex], in the formula for the perimeter of a rectangle:

[latex]{p}=2\left({l}\right)+2\left({w}\right)[/latex].

Rates

There is often a well-known formula or relationship that applies to a word problem. For example, if you were to plan a road trip, you would want to know how long it would take you to reach your destination. [latex]d=rt[/latex] is a well-known relationship that associates distance traveled, the rate at which you travel, and how long the travel takes.

Distance, Rate, and Time

If you know two of the quantities in the relationship [latex]d=rt[/latex], you can easily find the third using methods for solving linear equations. For example, if you know that you will be traveling on a road with a speed limit of [latex]30\frac{\text{ miles }}{\text{ hour }}[/latex] for 2 hours, you can find the distance you would travel by multiplying rate times time or [latex]\left(30\frac{\text{ miles }}{\text{ hour }}\right)\left(2\text{ hours }\right)=60\text{ miles }[/latex].

We can generalize this idea depending on what information we are given and what we are looking for. For example, if we need to find time, we could solve the [latex]d=rt[/latex] equation for t using division:

[latex]d=rt\\\frac{d}{r}=t[/latex]

Likewise, if we want to find rate, we can isolate r using division:

[latex]d=rt\\\frac{d}{t}=r[/latex]

In the following example you will see how this formula is applied to answer a few questions about ultra marathon running.

Ann Trason

Ann Trason

Ultra marathon running (defined as anything longer than 26.2 miles) is becoming very popular among women even though it remains a male-dominated niche sport. Ann Trason has broken twenty world records in her career. One such record was the American River 50-mile Endurance Run which begins in Sacramento, California, and ends in Auburn, California.[1] In 1993 Trason finished the run with a time of 6:09:08.  The men’s record for the same course was set in 1994 by Tom Johnson who finished the course with a time of 5:33:21.[2]

In the next examples we will use the [latex]d=rt[/latex] formula to answer the following questions about the two runners.

  1. What was each runner’s rate for their record-setting runs?
  2. What was each runner’s time for running one mile?

To make answering the questions easier, we will round the two runners’ times to 6 hours and 5.5 hours.

Example 2

What was each runner’s rate for their record-setting runs?

Now we will tackle the second question where we are asked to find a time for each runner.

Example 3

What was each runner’s time for running one mile?

In the following video, we show another example of answering many rate questions given distance and time.

Area

The area of a triangle is given by [latex] A=\frac{1}{2}bh[/latex] where

[latex]A[/latex] = area
[latex]b[/latex] = the length of the base
[latex]h[/latex] = the height of the triangle

Remember that when two variables or a number and a variable are sitting next to each other without a mathematical operator between them, you can assume they are being multiplied. This can seem frustrating, but you can think of it like mathematical slang. Over the years, people who use math frequently have just made that shortcut enough that it has been adopted as convention.

In the next example we will use the formula for area of a triangle to find a missing dimension, as well as use substitution to solve for the base of a triangle given the area and height.

Example 4

Find the base, [latex]b[/latex], of a triangle with an area, [latex]A[/latex], of 20 square feet and a height, [latex]h[/latex], of 8 feet.

We can rewrite the formula in terms of b or h as we did with perimeter previously. This probably seems abstract, but it can help you develop your equation-solving skills, as well as help you get more comfortable with working with all kinds of variables, not just x.

Example 5

Use the multiplication and division properties of equality to isolate the variable [latex]b[/latex] in the Area formula [latex] A=\frac{1}{2}bh[/latex].


Use the multiplication and division properties of equality to isolate the variable [latex]h[/latex] in the Area formula [latex] A=\frac{1}{2}bh[/latex].

The following video shows another example of finding the base of a triangle given area and height.

Temperature

Let’s look at another formula that includes parentheses and fractions, the formula for converting from the Fahrenheit temperature scale to the Celsius scale.

[latex]C=\left(F-32\right)\cdot \frac{5}{9}[/latex]

This formula is telling us that to find the equivalent Celsius temperature, we will start with the Fahrenheit temperature, subtract 32 from it, and then multiply the result by [latex]\frac{5}{9}[/latex]. But what if we are starting with a Celsius temperature and want to convert to Fahrenheit? The example below will show how you can use the same formula to convert from Celsius to Fahrenheit.

Example 6

Given a temperature of [latex]12^{\circ}{C}[/latex], find the equivalent in [latex]{}^{\circ}{F}[/latex].

In the next video, we show another example of converting from celsius to fahrenheit.

As with the other formulas we have worked with, in the examples above, we could have isolated the variable F first, then substituted in the given temperature in Celsius. The example below will show you how to solve the formula for F.

Example 7

The formula below is used for converting from the Fahrenheit scale to the Celsius scale. Solve the formula for [latex]F[/latex]. (This will change it into a formula that can be used to convert Celsius to Fahrenheit).

[latex]C=\left(F-32\right)\cdot \frac{5}{9}[/latex]

Think About It

Express the formula for the surface area of a cylinder, [latex]s=2\pi rh+2\pi r^{2}[/latex], in terms of the height, [latex]h[/latex].

In this example, the variable [latex]h[/latex] is buried pretty deeply in the formula for surface area of a cylinder. Using the order of operations, it can be isolated. Before you look at the solution, use the box below to write down what you think is the best first step to take to isolate [latex]h[/latex].


  1. "Ann Trason." Wikipedia. Accessed May 05, 2016. https://en.wikipedia.org/wiki/Ann_Trason.
  2.  "American River 50 Mile Endurance Run." Wikipedia. Accessed May 05, 2016. https://en.wikipedia.org/wiki/American_River_50_Mile_Endurance_Run.