### Learning OUTCOMES

- Solve distance, rate, and time problems
- Solve area, volume, and perimeter problems
- Solve temperature conversion problems
- Rearrange formulas to isolate specific variables
- Identify an unknown given a formula

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the **area** of a rectangular region, [latex]A=LW[/latex]; the **perimeter** of a rectangle, [latex]P=2L+2W[/latex]; and the **volume** of a rectangular solid, [latex]V=LWH[/latex]. When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

## Distance, Rate, and Time

### Example

It takes Andrew [latex]30[/latex] min to drive to work in the morning. He drives home using the same route, but it takes [latex]10[/latex] min longer and he averages [latex]10 mi/h[/latex] less than in the morning. How far does Andrew drive to work?

In the next example, we will find the length and width of a rectangular field given its perimeter and a relationship between its sides.

### Example

The perimeter of a rectangular outdoor patio is [latex]54[/latex] ft. The length is [latex]3[/latex] ft greater than the width. What are the dimensions of the patio?

The following video shows an example of finding the dimensions of a rectangular field given its perimeter.

### Example

The perimeter of a tablet of graph paper is [latex]48[/latex] in. The length is [latex]6[/latex] in. more than the width. Find the area of the graph paper.

The following video shows another example of finding the area of a rectangle given its perimeter and the relationship between its side lengths.

## Volume

### Example

Find the dimensions of a shipping box given that the length is twice the width, the height is [latex]8[/latex] inches, and the volume is [latex]1,600[/latex] in.^{3}.

### 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, *h*.

In this example, the variable *h* 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 *h*.

In the following video we show how to find the volume of a right circular cylinder formed from a given rectangle.

## Isolate Variables in Formulas

Sometimes, it is easier to isolate the variable you are solving for when you are using a formula. This is especially helpful if you have to perform the same calculation repeatedly, or you are having a computer perform the calculation repeatedly. In the next examples, we will use algebraic properties to isolate a variable in a formula.

### Example

Isolate the term containing the variable *w *from the formula for the perimeter of a rectangle*: *

[latex]{P}=2\left({L}\right)+2\left({W}\right)[/latex].

### Example

Use the multiplication and division properties of equality to isolate the variable *b* given [latex]A=\dfrac{1}{2}\normalsize bh[/latex]

Use the multiplication and division properties of equality to isolate the variable *h *given [latex]A=\dfrac{1}{2}\normalsize bh[/latex]

## Temperature

Let us 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\dfrac{5}{9}[/latex]

### Example

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

As with the other formulas we have worked with, we could have isolated the variable F first then substituted in the given temperature in Celsius.

### Example

Solve the formula shown below for converting from the Fahrenheit scale to the Celsius scale for F.

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