Problem 6 - Precalculus Mathematics for Calculus

Question

The formula \(d=r t\) models the distance \(d\) traveled by an object moving at the constant rate \(r\) in time \(t .\) Find formulas for the following quantities. $$r= \text {____} \qquad t = \text {____} .$$

Step-by-Step Solution

Answer

\(r = \frac{d}{t}\) and \(t = \frac{d}{r}\).

Step 1: Understand the Given Formula

The formula given is \(d = rt\), where \(d\) is the distance traveled, \(r\) is the rate of travel, and \(t\) is the time taken to travel that distance. Our goal is to find expressions for \(r\) and \(t\).

Step 2: Solve for Rate \(r\)

To find the formula for the rate \(r\), we need to rearrange the formula \(d = rt\) by isolating \(r\). Divide both sides of the equation by \(t\): \\[ r = \frac{d}{t} \] \This formula gives the rate \(r\) in terms of distance \(d\) and time \(t\).

Step 3: Solve for Time \(t\)

Next, we find the formula for the time \(t\) by isolating \(t\) in the original equation \(d = rt\). Divide both sides of the equation by \(r\): \\[ t = \frac{d}{r} \] \This formula gives the time \(t\) in terms of distance \(d\) and rate \(r\).

Key Concepts

Understanding Rate of Travel

The rate of travel is an essential concept that helps us understand how fast something is moving. In the context of the formula \(d = rt\), the rate of travel is represented by \(r\). It describes the speed at which an object moves over a certain distance. Rate of travel is typically measured in units such as miles per hour (mph) or kilometers per hour (kph).

Understanding this concept allows us to solve problems related to motion, such as finding out how fast a car needs to go to reach a destination within a specific time.

• If you have the distance \(d\) and the time \(t\), you can find the rate \(r\) using the formula \(r = \frac{d}{t}\).
• This tells you the average speed of the object across the given distance.

Knowing how to determine the rate of travel is key to solving many real-life problems involving movement.

Time Calculation

Time calculation is a fundamental aspect when dealing with motion. It's often crucial to figure out how long it will take to travel a particular distance at a given rate. With the formula \(d = rt\), rearranging it to find time \(t\) gives us \[ t = \frac{d}{r} \].

This equation is handy when you have a set distance and need to determine how much time you'll need when moving at a specific rate. Here are some points to consider:

• To find the time, divide the total distance by the rate of travel.
• If the distance is 100 miles and the rate is 50 mph, the time taken will be \(t = \frac{100}{50} = 2\) hours.
• Pay attention to the units to ensure they match (like miles with hours for mph).

Time calculation helps in planning, whether it's preparing travel itineraries or managing time efficiently in daily activities.

Rearranging Equations

Rearranging equations is a valuable skill in mathematics as it allows you to solve for different variables in a formula. With the distance formula \(d = rt\), knowing how to manipulate it gives you the ability to find either rate \(r\) or time \(t\), depending on what is needed.

Here are the steps to rearrange the equation:

• To find rate \(r\), divide both sides by \(t\) to get \(r = \frac{d}{t}\).
• To find time \(t\), divide both sides by \(r\) resulting in \(t = \frac{d}{r}\).

Rearranging equations is not only about plugging in values but understanding the relationships between variables.
This comprehension allows you to apply the formula flexibly in a range of problems, ensuring you can adapt your calculations to fit different scenarios.