Problem 6 - Algebra and Trigonometry

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.

Step-by-Step Solution

Answer

Rate: \(r = \frac{d}{t}\); Time: \(t = \frac{d}{r}\).

Step 1: Understanding the Formula

The formula is given as \(d = rt\), where \(d\) represents the distance traveled, \(r\) represents the rate or speed, and \(t\) is the time. This formula shows that distance is the product of rate and time.

Step 2: Derive Formula for Rate \(r\)

To find the formula for the rate, rearrange the formula \(d = rt\) to solve for \(r\). Divide both sides by \(t\) to isolate \(r\):\[r = \frac{d}{t}\].

Step 3: Derive Formula for Time \(t\)

To find the formula for time, rearrange the formula \(d = rt\) to solve for \(t\). Divide both sides by \(r\) to isolate \(t\):\[t = \frac{d}{r}\].

Key Concepts

Distance

The concept of distance is a fundamental idea in understanding movement. It is generally defined as how far an object has traveled over a period of time. In mathematical terms, this can be calculated using the formula \(d = rt\), where \(d\) represents distance, \(r\) is the rate, and \(t\) is the time. This equation shows that distance is directly proportional to both the rate and the time:

• More distance is covered with increased rate and time.
• If either rate or time is zero, the distance is zero, as an object at rest does not move.

Understanding distance in this equation means recognizing that it's a measure of the path taken by an object from a starting point to an endpoint. It allows you to quantify how much ground an object has covered due to its movement over time.

Rate

To grasp the concept of rate, envision it as the speed or velocity at which something moves. Rate tells us how fast an object is traveling over a given amount of time. It is a crucial element in the distance formula \(d = rt\). If you rearrange this formula to solve for rate, you get \(r = \frac{d}{t}\). This equation implies:

• The rate is equal to the distance traveled divided by the time taken.
• A higher rate signifies faster movement over the same time period, leading to more distance covered.
• If distance remains constant, an increase in rate will decrease the time needed to cover the distance.

Rate is commonly expressed in units such as miles per hour (mph) or kilometers per hour (km/h), making it easier to relate to everyday experiences such as driving a car or running.

Time

Time, in the context of the distance formula, denotes the duration over which an object is traveling. It plays a significant role in determining both distance and rate. When considering the formula \(d = rt\), you can derive the formula for time by rearranging it to \(t = \frac{d}{r}\).Key insights from this equation include:

• Time is the result of dividing the distance by the rate.
• More time is needed to cover a distance if the rate is slower.
• For a fixed distance, increasing the rate will reduce the time needed.

Time is typically measured in units such as seconds, minutes, or hours. Having a precise understanding of time's role in calculating distance and rate allows for practical real-life applications, like estimating trip durations and scheduling activities.