## Problems Involving Formulas II

### Learning Outcomes

• Solve a formula for a specific variables
• Evaluate a formula using substitution
• Solve area and perimeter words problems
• Solve temperature conversion problems

Formulas come up in many different areas of life. We have seen the formula that relates distance, rate, and time, and the formula for simple interest on an investment. In this section, we will look further at formulas, and see examples of formulas for dimensions of geometric shapes, as well as the formula for converting temperature between Fahrenheit and Celsius.

## Geometry

There are many geometric shapes that have been well studied over the years. We know quite a bit about circles, rectangles, and triangles. Mathematicians have proven many formulas that describe the dimensions of geometric shapes including area, perimeter, surface area, and volume.

### Perimeter

Perimeter is the distance around an object. For example, consider a rectangle with a length of $8$ and a width of $3$. There are two lengths and two widths in a rectangle (opposite sides), so we add $8+8+3+3=22$. Since there are two lengths and two widths in a rectangle, you may find the perimeter of a rectangle using the formula ${P}=2\left({L}\right)+2\left({W}\right)$ where

$L$ = Length

$W$ = Width

In the following example, we will use the problem-solving method we developed to find an unknown width using the formula for the perimeter of a rectangle. By substituting the dimensions we know into the formula, we will be able to isolate the unknown width and find our solution.

### Example

You want to make another garden box the same size as the one you already have. You write down the dimensions of the box and go to the lumber store to buy some boards. When you get there, you realize you didn’t write down the width dimension—only the perimeter and length. You want the exact dimensions so you can have the store cut the lumber for you.

Here is what you have written down:

Perimeter = $16.4$ feet
Length = $4.7$ feet

Can you find the dimensions you need to have your boards cut at the lumber store? If so, how many boards do you need and what lengths should they be?

This video shows a similar garden box problem.

We could have isolated the $W$ in the formula for perimeter before we solved the equation, and if we were going to use the formula many times, it could save a lot of time. The next example shows how to isolate a variable in a formula before substituting known dimensions or values into the formula.

### Example

Isolate the term containing the variable, $W$, from the formula for the perimeter of a rectangle:

${P}=2\left({L}\right)+2\left({W}\right)$.

## Area

The area of a triangle is given by $A=\frac{1}{2}bh$ where

$A$ = area
$b$ = the length of the base
$h$ = the height of the triangle

Remember, that when two variables, or a number and a variable, are sitting next to each other without a mathematical operator between them, you can assume they are being multiplied. This can seem frustrating, but you can think of it like mathematical slang. Over the years, people who use math frequently have just made that shortcut enough times that it has been adopted as convention.

In the next example, we will use the formula for area of a triangle to find a missing dimension, as well as use substitution to solve for the base of a triangle given the area and height.

### Example

Find the base ($b$) of a triangle with an area ($A$) of $20$ square feet and a height ($h$) of $8$ feet.

We can rewrite the formula in terms of $b$ or $h$ as we did with the perimeter formula previously. This probably seems abstract, but it can help you develop your equation-solving skills, as well as help you get more comfortable with working with all kinds of variables, not just $x$.

### Example

1. Use the multiplication and division properties of equality to isolate the variable $b$.

2. Use the multiplication and division properties of equality to isolate the variable $h$.

The following video shows another example of finding the base of a triangle, given area and height.

Express the formula for the surface area of a cylinder, $S=2\pi rh+2\pi r^{2}$, in terms of the height, $h$.

In this example, the variable $h$ is buried pretty deep in the formula for the 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 $h$.

## Temperature

Let’s look at another formula that includes parentheses and fractions: the formula for converting from the Fahrenheit temperature scale to the Celsius scale.

$C=\left(F-32\right)\cdot \frac{5}{9}$

### Example

Given a temperature of $12^{\circ}{C}$, find the equivalent in ${}^{\circ}{F}$.

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 $F$.

$C=\left(F-32\right)\cdot \frac{5}{9}$

In the last video, we show how to convert from Celsius to Fahrenheit.

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