Using the Distance Formula to Solve Word Problems

Use the Distance, Rate, and Time Formula

Read about the distance, rate, and time formula. We use this equation for all uniform motion problems. Then, do examples 2.58 through 2.60 and check your answers. If you want more practice, you can try exercises 376 and 387.

One formula you will use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant rate. Rate is an equivalent word for "speed". The basic idea of rate may already familiar to you. Do you know what distance you travel if you drive at a steady rate of 60 miles per hour for 2 hours? (This might happen if you use your car's cruise control while driving on the highway.) If you said 120 miles, you already know how to use this formula!

Distance, Rate, and Time

For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula:

\(\begin{aligned} d=r t \quad \text { where } \quad d &=\text { distance } \\ & r=\text { rate } \\ & t=\text { time } \end{aligned}\)

We will use the Strategy for Solving Applications that we used earlier in this chapter. When our problem requires a formula, we change Step 4. In place of writing a sentence, we write the appropriate formula. We write the revised steps here for reference.

HOW TO

Solve an application (with a formula).

Step 1. Read the problem. Make sure all the words and ideas are understood.
Step 2. Identify what we are looking for.
Step 3. Name what we are looking for. Choose a variable to represent that quantity.
Step 4. Translate into an equation. Write the appropriate formula for the situation. Substitute in the given information.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. Answer the question with a complete sentence.

You may want to create a mini-chart to summarize the information in the problem. See the chart in this first example.

Example 2.58

Jamal rides his bike at a uniform rate of 12 miles per hour for \(3\frac{1}{2}\) hours. What distance has he traveled?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.		distance traveled
Step 3. Name. Choose a variable to represent it.		Let \(d = distance\).
Step 4. Translate: Write the appropriate formula.		\(d=rt\)
		\(d=?\) \(r=12 \mathrm{mph}\) \(t=3 \frac{1}{2}\) hours
Substitute in the given information.		\(d=12 \cdot 3 \frac{1}{2}\)
Step 5. Solve the equation.		\( d\)=42 miles
Step 6. Check
Does 42 miles make sense?
Jamal rides:

Step 7. Answer the question with a complete sentence.		Jamal rode 42 miles.

Try It 2.115

Lindsay drove for \(5 \frac{1}{2}\) hours at 60 miles per hour. How much distance did she travel?

Try It 2.116

Trinh walked for \(2 \frac{1}{3}\) hours at 3 miles per hour. How far did she walk?

Example 2.59

Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of 520 miles. If he can drive at a steady rate of 65 miles per hour, how many hours will the trip take?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.		How many hours (time)
Step 3. Name. Choose a variable to represent it.		Let \(t = time\)..
		\(d=520 \, \text{miles}\) \(r=65 \, \text{mph}\) \(t=? \, \text{hours}\)
Step 4. Translate. Write the appropriate formula.		\(d=rt\)
Substitute in the given information.		\(520=65t\)
Step 5. Solve the equation.		\(t=8\)
Step 6. Check. Substitute the numbers into the formula and make sure the result is a true statement.
	\(\begin{aligned} d &=r t \\ 520 & \stackrel{?}{=} 65 \cdot 8 \\ 520 &=520 \text{✓} \end{aligned}\)
Step 7. Answer the question with a complete sentence. Rey's trip will take 8 hours.

Try It 2.117

Lee wants to drive from Phoenix to his brother's apartment in San Francisco, a distance of 770 miles. If he drives at a steady rate of 70 miles per hour, how many hours will the trip take?

Try It 2.118

Yesenia is 168 miles from Chicago. If she needs to be in Chicago in 3 hours, at what rate does she need to drive?

Source: OpenStax, https://openstax.org/books/elementary-algebra/pages/2-6-solve-a-formula-for-a-specific-variable
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