# Math Models and Geometry

## Solve a Formula for a Specific Variable


### Use the Distance, Rate, and Time Formula

One formula you’ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you travel if you drove at a steady rate of  miles per hour for  hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said  miles, you already know how to use this formula!
The math to calculate the distance might look like this:
In general, the formula relating distance, rate, and time is
Notice that the units we used above for the rate were miles per hour, which we can write as a ratio  Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles.

### Solve a Formula for a Specific Variable

In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft ExcelTM relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.
In  and , we used the formula  This formula gives the value of  when you substitute in the values of  and  But in , we had to find the value of  We substituted in values of  and  and then used algebra to solve to  If you had to do this often, you might wonder why there isn’t a formula that gives the value of  when you substitute in the values of  and  We can get a formula like this by solving the formula  for 
To solve a formula for a specific variable means to get that variable by itself with a coefficient of  on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable in general. This process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.
Let’s try a few examples, starting with the distance, rate, and time formula we used above.
In Solve Simple Interest Applications, we used the formula  to calculate simple interest, where  is interest,  is principal,  is rate as a decimal, and  is time in years.
Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually  and  You might be given an equation that is solved for  and need to solve it for  or vice versa. In the following example, we’re given an equation with both  and  on the same side and we’ll solve it for  To do this, we will follow the same steps that we used to solve a formula for a specific variable.
In the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you’re ready to solve a formula in general without using numbers as a guide?

### Key Concepts

1. Distance, Rate, and Time
  

### Section Exercises


### Practice Makes Perfect

Use the Distance, Rate, and Time Formula 
In the following exercises, solve.
Solve a Formula for a Specific Variable 
In the following exercises, use the formula. 
In the following exercises, use the formula 
In the following exercises, use the formula 
In the following exercises, solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


### Chapter Review Exercises


### Use a Problem Solving Strategy

Approach Word Problems with a Positive Attitude 
In the following exercises, solve.
Use a Problem Solving Strategy for Word Problems 
In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.
Solve Number Problems 
In the following exercises, solve each number word problem.

### Solve Money Applications

Solve Coin Word Problems 
In the following exercises, solve each coin word problem.
Solve Ticket and Stamp Word Problems 
In the following exercises, solve each ticket or stamp word problem.

### Use Properties of Angles, Triangles, and the Pythagorean Theorem

Use Properties of Angles 
In the following exercises, solve using properties of angles.
Use Properties of Triangles 
In the following exercises, solve using properties of triangles.
In the following exercises,  is similar to  Find the length of the indicated side.
Use the Pythagorean Theorem 
In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.
In the following exercises, solve. Approximate to the nearest tenth, if necessary.

### Use Properties of Rectangles, Triangles, and Trapezoids

Understand Linear, Square, Cubic Measure 
In the following exercises, would you measure each item using linear, square, or cubic measure?
In the following exercises, find 
Use Properties of Rectangles 
In the following exercises, find the ⓐ perimeter ⓑ area of each rectangle 
In the following exercises, solve.
Use Properties of Triangles 
In the following exercises, solve using the properties of triangles.
Use Properties of Trapezoids 
In the following exercises, solve using the properties of trapezoids.

### Solve Geometry Applications: Circles and Irregular Figures

Use Properties of Circles 
In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth.
Find the Area of Irregular Figures 
In the following exercises, find the area of each shaded region.

### Solve Geometry Applications: Volume and Surface Area

Find Volume and Surface Area of Rectangular Solids 
In the following exercises, find the 
Find Volume and Surface Area of Spheres 
In the following exercises, find the
Find Volume and Surface Area of Cylinders 
In the following exercises, find the 
Find Volume of Cones 
In the following exercises, find the volume of the cone.

### Solve a Formula for a Specific Variable

Use the Distance, Rate, and Time Formula 
In the following exercises, solve using the formula for distance, rate, and time.
Solve a Formula for a Specific Variable 
In the following exercises, use the formula 
In the following exercises, use the formula 
In the following exercises, use the formula 
In the following exercises, solve.

### Chapter Practice Test
