Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

A car is stationary at a toll booth. Eighteen minutes later at a point 20 miles down the road the car is clocked at 60 miles per hour. Sketch a possible graph of \(v\) versus \(t\). Sketch a possible graph of the distance traveled \(s\) against \(t .\) Use the Mean Value Theorem to show that the car must have exceeded the 60 mile per hour speed limit at some time after leaving the toll booth, but before the car was clocked at 60 miles per hour.

Step-by-Step Solution

Answer

The car must have exceeded 66.67 mph before being clocked at 60 mph.

Step 1: Understand the Problem

The car travels 20 miles in 18 minutes and is later observed traveling at 60 mph. We need to determine if the car exceeded 60 mph before being clocked.

Step 2: Mean Value Theorem Application

To use the Mean Value Theorem, recall that it applies to continuous functions. \[ s'(t) = \frac{s(b) - s(a)}{b - a} \]In this context, \(s(t)\) is the distance function and \(t\) is time. We convert 18 minutes to hours: \( t = \frac{18}{60} = 0.3 \) hours. Over 0.3 hours, the car traveled 20 miles, so,\[ \frac{s(0.3) - s(0)}{0.3 - 0} = \frac{20}{0.3} \approx 66.67 \text{ mph} \]This suggests at some point, the instantaneous speed \(s'(t)\) must have been 66.67 mph, exceeding 60 mph.

Step 3: Graph Sketch of Velocity vs Time

The graph of velocity \(v(t)\) starts at 0 mph (as the car is stationary), increases possibly above 60 mph due to acceleration, and is then 60 mph after 18 minutes (0.3 hours). Thus, the velocity graph is likely increasing above 60 mph before stabilizing at 60 mph.

Step 4: Graph Sketch of Distance vs Time

Distance \(s(t)\) increases from 0 as a non-linear curve beginning steeply (because velocity increases) and then becomes a linear line after reaching 60 mph. The distance \(s\) over 0.3 hours must reach 20 miles.

Key Concepts

velocity-time graph

A velocity-time graph provides a visual representation of the car's speed, or velocity, over a period of time. Initially, the car is at rest, starting at 0 mph. As time progresses, it accelerates. Each point on the graph corresponds to a specific velocity at a given time.

The graph typically starts at zero, showing how the car's velocity increases from being stationary. As it accelerates, the line will slope upwards, possibly exceeding the speed limit of 60 mph temporarily. This represents the period where the car gained speed to reach 20 miles in 18 minutes.

Eventually, the graph will level out to 60 mph at 0.3 hours, the time when the car is clocked. This flattening indicates a stable velocity. This transition from acceleration to a constant velocity can highlight when the car exceeded the speed limit if analyzed closely using the Mean Value Theorem.

distance-time graph

The distance-time graph tracks the ground covered by the car over time. At the start, the car is at the toll booth, so distance is zero. As the car moves, the graph will show an upward trend, indicating the continuous increase in distance traveled.

Since the car accelerates initially, the graph starts with a steeper curve. In this phase, the line represents a nonlinear increase in distance, as the speed is not constant. Afterwards, once the car reaches the speed of 60 mph, the curve transitions to a linear trend.

• The steep initial section captures the rapid change in speed.
• The subsequent linear part shows constant speed with distance growing steadily over time, exactly mirroring the car's movement at 60 mph.

Ultimately, after 0.3 hours, the total distance of 20 miles is marked on the graph.

instantaneous speed

Instantaneous speed refers to the exact speed of the car at a particular instant. Unlike average speed, it does not cover the entire journey but focuses on a point in time. It can vary greatly during acceleration or deceleration phases.

In our scenario, at some point during the journey, the car's instantaneous speed must have exceeded 60 mph. This is because the car covered 20 miles in only 0.3 hours. This is supported by the Mean Value Theorem, which implies that there must be at least one moment where the instantaneous speed was higher than the average speed over that interval.

By examining the velocity-time graph, we can see where the speed might have been even higher than when it stabilized at 60 mph. Such analysis is vital in understanding variations in speed during a trip.

average speed

Average speed is calculated by dividing the total distance traveled by the total time taken. It provides a simple measure of how fast, in general, the car moved over its journey.

In the example of the car journey, the average speed can be found by taking the total distance of 20 miles and dividing it by the time of 0.3 hours:\[\text{Average Speed} = \frac{20}{0.3} = 66.67 \text{ mph}\]

This shows that, on average, the car moved faster than the speed limit. The average calculates the entire journey without detailing any specific points where the speed might have varied as analyzed from instantaneous speed. Understanding both average and instantaneous speeds helps in assessing the journey's dynamics effectively.