## 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 $2x+1=7$ has the unique solution $x=3$ because when we substitute 3 for $x$ in the equation, we obtain the true statement $2\left(3\right)+1=7$.

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 $A$ of a circle in terms of the radius $r$ of the circle: $A=\pi {r}^{2}$. For any value of $r$, the area $A$ can be found by evaluating the expression $\pi {r}^{2}$.

### Example 11: Using a Formula

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

Figure 3. Right circular cylinder

### Solution

Evaluate the expression $2\pi r\left(r+h\right)$ for $r=6$ and $h=9$.

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

The surface area is $180\pi$ square inches.

### Try It 11

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 $A=\left(L+16\right)\left(W+16\right)-L\cdot W$. Find the area of a matte for a photograph with length 32 cm and width 24 cm.

Solution