Problem 70 - Algebra and Trigonometry

Question

Jason and Debbie leave Detroit at \(2 : 00\) P.M. and drive at a constant speed, traveling west on \(\mathrm{I}-90 .\) They pass Ann Arbor, 40 \(\mathrm{mi}\) from Detroit, at \(2 : 50 \mathrm{P} \mathrm{M}\) (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent?

Step-by-Step Solution

Answer

(a) Distance traveled: \( d(t) = 48t \). (b) Graph is a line with positive slope 48. (c) Slope is 48, representing speed in mph.

Step 1: Determine Time Elapsed

Identify the time it took Jason and Debbie to travel from Detroit to Ann Arbor. Since they left at 2:00 PM and passed Ann Arbor at 2:50 PM, the time taken is 50 minutes, which we convert to hours: \(50 \text{ minutes} = \frac{50}{60} \text{ hours} \approx 0.833\text{ hours}.\)

Step 2: Calculate the Speed

The speed can be calculated by dividing the distance by the time. They traveled 40 miles in approximately 0.833 hours, so their speed is \( \frac{40}{0.833} \text{ miles per hour} \approx 48 \text{ mph}.\)

Step 3: Express Distance as a Function of Time

Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), we express the distance \( d(t) \) as a function of time \( t \) in hours since 2:00 PM: \( d(t) = 48t \).

Step 4: Draw the Graph of Distance vs. Time

To graph the equation \( d(t) = 48t \), plot time (in hours) on the x-axis and distance (in miles) on the y-axis. The graph should be a straight line starting at the origin (0,0) because they started at time 0 and distance 0.

Step 5: Determine Slope of the Line

The equation \( d(t) = 48t \) is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m = 48 \), which represents the speed of the travel, i.e., 48 miles per hour.

Key Concepts

Constant Speed Travel

Constant speed travel is a vital concept in understanding how distance and time interrelate during a journey. When someone travels at a constant speed, the rate of movement does not change. This translates into a linear relationship between time and distance. In the example of Jason and Debbie's travel from Detroit to Ann Arbor, they maintained a speed of 48 miles per hour.

The formula for calculating distance in this scenario is given by the equation:

• Distance = Speed × Time

This means that for every hour of travel, they consistently cover 48 miles. Calculating this in real time, it's observed that in 50 minutes, which translates to approximately 0.833 hours, they traveled 40 miles. Maintaining a constant speed implies predictability and straightforward calculations, as both speed and time vary directly, enabling travelers to plan more effectively.

Slope Interpretation

The slope in a distance-time graph has a special meaning, particularly when interpreting travel scenarios. In the context of Jason and Debbie's journey, the slope of their distance-time graph represents their speed.

The slope formula in mathematics, expressed as:

• Slope (m) = Change in Distance / Change in Time

is used to determine the rate of speed in this case. Calculating from their trip details, the slope comes out to be 48, which directly corresponds to their travel speed: 48 miles per hour.

Understanding slope is crucial in these types of problems as it quantifies how quickly distance changes over time. It allows for an easy interpretation of how fast someone is traveling without having to manually calculate speed for every checkpoint or mile.

Graphing Linear Functions

Graphing linear functions is an excellent way to visually communicate the relationship between two variables—in this case, time and distance. The linear equation derived from Jason and Debbie’s travel, represented as:

• Distance = 48t

is a perfect example of a linear function, where the distance varies linearly with time.

To create the graph:

• Place time on the x-axis.
• Place distance on the y-axis.
• Plot the line that passes through the origin (0,0).

This straight line clearly shows that as time increases, distance increases at a constant rate— without any curves or bends. Graphing such linear functions is a useful skill as it provides a clear and immediate way to interpret many real-world scenarios where two factors increase together proportionally.