A. Using Formulas

A formula is a statement that claims that two expressions are equal. When dealing with formulas, it is important to know what each variable represents. Generally, formulas are casesensitive, which means that an uppercase may represent a different variable than a lowercase . If more than one variable from the same type of variables is needed, then subscripts are often used.

Here are three examples of formulas.

In most formulas, the value of the variable on the left is found by substituting values for the variable(s) on the right.

The following three steps occur when evaluating a formula to determine the numerical value of one of the variables:

Next, we consider some examples involving finding the mean, slope, and total amount.

Example 1. Find the mean of the numbers 2, , and 5.

Example 2. Calculate the slope of the line that passes through () and (2, 5).

(Notice that subtracting a negative number gives the same result as adding its opposite: .)

Example 3. Suppose that you invest $1000 in an account that pays 4% interest on the initial amount invested. How much money is in your account after 5 years?

The amount of money in the account after 5 years is .

Sometimes when working with formulas, the variable for which you want to solve is not isolated on the left-hand side of the equation. In that case, additional algebraic manipulation is required to solve for the unknown variable. The following two common algebraic manipulations will often do the trick:

Example 4. Suppose you are told that the mean of four numbers is 10, and you know that the first three numbers are 8, 12, and 6. What is the value of the fourth number?

We have found that the fourth number must be 14.

Example 5. The slope of a line through two points is %. The first point has coordinates (1, 1). The -coordinate of the second point is 3. What is the x-coordinate of the second point?

Calculator note: When calculating the value of a fraction, such as when you find the slope of a line given two points on the line, you must separately enclose both the numerator and denominator in parentheses.

Example 6. Using a calculator, find the slope of a line that passes through the points (5, 9) and (17, 6).

The keystrokes for a TI-84 graphing calculator would be:

which yields .

This is how this calculation should appear on your calculator screen.

Practice Exercises

Use the formulas above and the three-step method to find the answers to Practice Exercises 1–3.

B. solving for one Variable in Terms of Another

Sometimes when working with formulas, the variable of interest is not isolated on the left-hand side of the equation. In that case, additional algebraic manipulation is required to solve for the variable of interest in terms of the remaining variables. The following two common algebraic manipulations will often do the trick:

For example, is the formula for finding the circumference of a circle, , given its diameter, . However, in the context of forestry, a forester would measure a tree’s circumference in order to calculate its diameter. So, the forester would prefer having a formula that gives in terms of . Dividing both sides of this formula by gives

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Example 1. Solve for .

If the equation involves a single radical expression involving the variable of interest, a different approach is needed. In that case, first isolate the radical on one side of the equation. Raise both sides of the equation to the power required to remove the radical. Finally, use the algebraic manipulations in the two bullets above to solve for the variable of interest in terms of the other variables. [For information on radicals and powers, see Algebra Review VI, item A, Powers and Roots (page AR-22), and item D, Rules for Exponents and Roots (page AR-25).]

Example 2. Solve for .

You can use the same technique used in Example 2 if the variable of interest is raised to a power.

Example 3. Solve .

Practice Exercises

Use the simple interest formula (from Chapter 21, page 871), , for Practice Exercises 1 and 2.

Use the compound interest formula (from Chapter 21, page 875), , for Practice Exercises 3 and 4.

C. Formulas Related to Geometric shapes

In Chapters 18 and 20, you will encounter a variety of twodimensional and three-dimensional geometric shapes. Formulas for perimeter and area are useful when dealing with two-dimensional shapes. Formulas for surface area and volume are useful when dealing with three-dimensional shapes. This section focuses on a few of the most commonly used two- and three-dimensional geometric shapes.

Two-Dimensional Shapes

Note: Squares are special cases of rectangles, where .

Note: For circles, the length around the circle is called the circumference rather than the perimeter.

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Example 1. Jordon plans to fence in a 12-ft-by-8-ft rectangular enclosure for her dog. Determine the length of fencing needed to make the enclosure and the area that will be enclosed.

The perimeter gives the length of fencing that is needed: . The enclosed area is .

Three-Dimensional Shapes

Example 2. For a party, Cara made paper cones to hold candy and popcorn (see diagram).

For an open cone, we need to remove the area of the circular top. The area of a circle with a 3-in. radius is . The surface area is approximately , or

Practice Exercises