# Formulas

An

A

The surface area is

A photograph with length

Solution

**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 $\pi$

.### Solution

Evaluate the expression$2\pi r\left(r+h\right)$

for $r=6$

and $h=9$

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

$180\pi$

square inches.### Try It 11

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 cm

^{2}) 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