Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Distance equals rate times time, \(d=r t\). Find the distance Tom travels if he is moving at a rate of \(55 \mathrm{mi} / \mathrm{h}\) for \(3.5 \mathrm{~h}\).

Step-by-Step Solution

Answer

Tom travels 192.5 miles.

Step 1: Identify the Formula and Variables

The given formula is \(d = rt\) where \(d\) is the distance, \(r\) is the rate or speed, and \(t\) is the time. We need to find the distance \(d\).

Step 2: Substitute Known Values into the Formula

Tom's rate \(r\) is given as \(55\) miles per hour and the time \(t\) is \(3.5\) hours. Substitute these values into the formula \(d = rt\).

Step 3: Calculate the Distance

Substitute the values into the formula: \(d = 55 \cdot 3.5\). Multiply these values to calculate the distance.

Step 4: Perform the Multiplication

Multiply \(55\) by \(3.5\) to obtain the distance: \[d = 55 \times 3.5 = 192.5\].

Step 5: State the Answer

The distance Tom travels is \(192.5\) miles. This is the result of multiplying the rate by the time.

Key Concepts

Rate and Time Calculation

In the world of mathematics and physics, understanding the relationship between distance, rate, and time is crucial. The formula that connects these concepts is \( d = rt \), where \(d\) stands for distance, \(r\) for rate (or speed), and \(t\) for time. This formula is fundamental for calculating how far an object travels when moving at a constant speed over a given time period. Let's explore how each component can be used:

• Rate: This is often known as speed in layperson terms. It's how fast something is moving. Typically, this is expressed in units like miles per hour (mph) or kilometers per hour (kph).
• Time: The duration that the object has been moving. This could be in seconds, minutes, or hours, depending on the context.
• Distance: Finally, this is the total length covered by the object. It is what you calculate when you know the rate and time.

To find the distance an object travels, you simply multiply the rate of travel by the time traveled. In our example, Tom travels at a rate of 55 mph for 3.5 hours. Using our formula, his distance traveled is \( d = 55 \times 3.5 \), resulting in 192.5 miles. This direct multiplication gives a clear picture of the distance based on the provided speed and time.

Algebraic Manipulation

Algebraic manipulation involves rearranging formulas and solving equations to extract useful information. With the equation \( d = rt \), we might occasionally want to solve for a different variable. In our scenario, we were interested in determining the distance \(d\), but if needed, we could solve for \(r\) or \(t\) instead, using basic algebraic steps.To rearrange the formula:

• If we need to solve for rate \(r\), divide both sides by time \(t\): \[ r = \frac{d}{t} \]
• To solve for time \(t\), divide both sides by rate \(r\): \[ t = \frac{d}{r} \]

These manipulations are handy in real-life problems, like determining how fast you need to go to reach a destination in a certain time, or how long it will take to arrive if you maintain a particular speed. Understanding how to juggle these variables is key to solving many practical problems you'll encounter.

Problem Solving in Algebra

Problem solving in algebra often involves identifying the variables and constants from the problem statement and plugging them into known formulas. In Tom's case, we started with the formula \( d = rt \) and plugged in the known values for \(r\) (55 mph) and \(t\) (3.5 hours).Effective problem solving requires:

• Understanding the Problem: Comprehend what the problem is asking. In our situation, it was asking for the distance traveled.
• Identifying Known vs. Unknown: Recognize what values you have (rate and time) and what you need to find (distance).
• Substituting Values: Place the known values into the formula and solve through the necessary calculations.
• Checking Your Work: Always re-evaluate your result to see if it logically makes sense.

Using these strategies simplifies the problem-solving process. Once you've grasped one part of the equation, apply the same methodology to solve for different scenarios or variables. Whether calculating distance, rate, or time, systematic problem solving ensures accuracy and builds confidence in using algebra to tackle a variety of questions.