Question

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

Short answer

By the average speed interpretation, as the average speed (7.67 knots) is greater than the given value (7.5 knots), so, at some point during the journey, the ship's speed must have exceeded 7.5 knots.

Step-by-step solution

Calculate Average Speed

In order to calculate the average speed, use the formula average speed = total distance/total time. Here total distance = 184 sea miles and total time = 24 hours. So, average speed = 184/24 = 7.67 knots.

Compare Average Speed with Given Speed

Having calculated the average speed to be 7.67 knots, compare it with the given speed of 7.5 knots.

Justify the speed exceeding at some point

The average speed of 7.67 knots is greater than 7.5 knots. Hence, for the average speed to be 7.67 knots, the ship must have travelled at speeds greater than 7.67 knots at some points - therefore, certainly more than 7.5 knots at some points. This is how we justify that the speed exceeded 7.5 knots, at least at some point during the journey.

Key concepts

Calculus in Kinematics

Kinematics is a branch of mechanics that deals with the motion of objects without considering the forces causing the motion. Calculus, with its principles of derivatives and integrals, plays a pivotal role in kinematics, particularly when analyzing variable motion.

As for the trireme exercise, although it isn't explicitly a calculus-based problem, understanding calculus can help us comprehend why the trireme's speed had to exceed 7.5 knots at some points. In calculus terms, the average speed is akin to the mean value theorem for integrals. This theorem essentially tells us that there is at least one point where the instantaneous speed equals the average speed over a given interval. Since the trireme's average speed was calculated at 7.67 knots, using calculus reasoning, we know that the instantaneous speed must have exceeded 7.5 knots at some point(s) during the 24-hour period.

Mathematical Problem Solving

The process of reaching a solution in a mathematical problem can involve various steps such as understanding the problem, devising a plan, executing the plan and reviewing the solution. In our example concerning the trireme, we follow a structured approach to problem-solving.

First, we clearly understand the problem: calculating whether the trireme's speed exceeded 7.5 knots at any time. Next, we formulate our plan, which involves using the average speed formula. After computing the average, we compare it to 7.5 knots, conducting a logical analysis based on the known information. Lastly, we interpret the result to confirm the scenario in which the speed exceeded 7.5 knots. This structure provides a framework that fosters clear thinking, enabling effective and efficient problem-solving.

Average Speed Formula

The concept of average speed is central to understanding motion over a period of time. It is calculated as the total distance traveled divided by the total time taken. Mathematically, if the total distance is represented by d and the total time is t, the formula is expressed as:
\[ \text{Average speed} = \frac{d}{t} \].

Using this formula, we calculated the average speed of the trireme to be 7.67 knots. This value tells us the consistent speed at which the trireme would need to move to cover 184 sea miles in 24 hours. However, naturally, the speed during travel varies due to multiple factors like wind, currents, and human endurance. As the exercise improvement advice suggests, it's important to understand that while the average speed provides a valuable summary of a journey, it does not give any information about the variability of the speed throughout the trip. The calculation confirms that the trireme must have exceeded the average speed at certain points to achieve the overall average.