Problem 53 - University Calculus: Early Transcendentals

Question

Classical accounts tell us that a 170 -oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme's speed exceeded 7.5 knots (sea or nautical miles per hour).

Step-by-Step Solution

Answer

The trireme's average speed was 7.67 knots, which proves its speed exceeded 7.5 knots during the trip.

Step 1: Understand the Problem

We are given that a trireme covered 184 sea miles in 24 hours and need to determine if its speed exceeded 7.5 knots at some point during this journey. One knot is equivalent to one nautical mile per hour.

Step 2: Calculate Average Speed

Calculate the average speed of the trireme using the total distance covered and the time taken. The formula for average speed is given by:\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\]Insert the given values:\[\text{Average Speed} = \frac{184 \text{ miles}}{24 \text{ hours}} = 7.6667 \text{ knots}\]

Step 3: Apply the Mean Value Theorem for Speed

According to the Mean Value Theorem in continuous motion, there must have been at least one moment during the 24-hour journey when the instantaneous speed of the trireme equaled its average speed. Since the average speed was 7.6667 knots, which is greater than 7.5 knots, there must have been moments when the speed exceeded 7.5 knots.

Step 4: Conclude the Analysis

Since the average speed of the trireme was greater than 7.5 knots, by the Mean Value Theorem, its speed exceeded 7.5 knots at some point during its journey.

Key Concepts

Average Speed Calculation

When determining the average speed of an object in motion, you assess the entire journey it took. Average speed is essentially a summary of how fast something traveled over a period. To calculate it, you use the formula:

• Average Speed = \( \frac{\text{Total Distance}}{\text{Total Time}} \)

For the trireme, which covered 184 nautical miles in 24 hours, the average speed is calculated as follows:

• Average Speed = \( \frac{184 \text{ miles}}{24 \text{ hours}} = 7.6667 \text{ knots} \)

This means over the whole journey, the trireme traveled at an average speed slightly over 7.5 knots.
While this figure gives a general idea, it doesn’t tell us the speed at any specific moment along the way.

Instantaneous Speed

Instantaneous speed refers to how fast an object is moving at a specific moment in time. It differs from average speed because it's a snapshot rather than an overview.
If you think about driving a car, your instantaneous speed is what the speedometer shows at any given moment.
In the context of the Mean Value Theorem, it tells us that during a journey, your instantaneous speed must equal your average speed at least once. Applying this to the trireme, since its average speed was calculated to be 7.6667 knots:

• By the Mean Value Theorem, at least once during its journey, the trireme's instantaneous speed must have been 7.6667 knots.

Because this exceeds 7.5 knots, the theorem guarantees there were moments when its speed exceeded that threshold.

Nautical Miles

Understanding nautical miles is crucial when exploring speed in the context of sea travel. Unlike the standard mile, a nautical mile is based on the Earth's latitude and longitude coordinates.
It equates to one minute of latitude, making it particularly useful for navigation over water, a firm basis for sea and air distances.
For instance, one nautical mile equals approximately 1.1508 standard miles. In terms of speed, a knot is one nautical mile per hour. So, when discussing the trireme:

• To say it travels at 7.6667 knots means it covers 7.6667 nautical miles each hour during its average journey.

This unit of measurement helps in accurately understanding and calculating maritime and aeronautical speeds.