Formulas

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[/latex] has the unique solution [latex]x=3[/latex] because when we substitute 3 for [latex]x[/latex] in the equation, we obtain the true statement [latex]2\left(3\right)+1=7[/latex].

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[/latex] of a circle in terms of the radius [latex]r[/latex] of the circle: [latex]A=\pi {r}^{2}[/latex]. For any value of [latex]r[/latex], the area [latex]A[/latex] can be found by evaluating the expression [latex]\pi {r}^{2}[/latex].

Example 11: Using a Formula

A right circular cylinder with radius [latex]r[/latex] and height [latex]h[/latex] has the surface area [latex]S[/latex] (in square units) given by the formula [latex]S=2\pi r\left(r+h\right)[/latex]. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex]\pi[/latex].

A right circular cylinder with an arrow extending from the center of the top circle outward to the edge, labeled: r. Another arrow beside the image going from top to bottom, labeled: h.

Figure 3. Right circular cylinder

Solution

Evaluate the expression [latex]2\pi r\left(r+h\right)[/latex] for [latex]r=6[/latex] and [latex]h=9[/latex].

[latex]\begin{array}\text{ }S\hfill&=2\pi r\left(r+h\right) \\ \hfill& =2\pi\left(6\right)[\left(6\right)+\left(9\right)] \\ \hfill& =2\pi\left(6\right)\left(15\right) \\ \hfill& =180\pi\end{array}[/latex]

The surface area is [latex]180\pi[/latex] square inches.

Try It 11

/ An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.

Figure 4

A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm2) is found to be [latex]A=\left(L+16\right)\left(W+16\right)-L\cdot W[/latex]. Find the area of a matte for a photograph with length 32 cm and width 24 cm.

Solution