Question

Multiple Choice Which of the following quantities would not be represented by the definite integral \(\int_{0}^{8} 70 d t ?\) (A) The distance traveled by a train moving at 70 mph for 8 minutes. (B) The volume of ice cream produced by a machine making 70 gallons per hour for 8 hours. (C) The length of a track left by a snail traveling at 70 \(\mathrm{cm}\) per hour for 8 hours. (D) The total sales of a company selling \(\$ 70\) of merchandise per hour for 8 hours. (E) The amount the tide has risen 8 minutes after low tide if it rises at a rate of 70 \(\mathrm{mm}\) per minute during that period.

Short answer

The correct choice is (E). The amount the tide has risen 8 minutes after low tide if it rises at a rate of 70 mm per minute during that period cannot be represented by the given integral, because the time units are minutes, not hours.

Step-by-step solution

Analyze the Options

For each choice, consider whether it could be represented by the given integral by checking if it involves a constant rate of 70 units per hour over 8 hours or not.

Evaluate Answer A

The train is moving at a constant rate of 70 mph, and we are considering an 8 hour period, so this answer could indeed be represented by the given integral.

Evaluate Answer B

Similarly, the ice cream machine is producing ice cream at a constant rate of 70 gallons per hour for 8 hours, so this answer could also be represented by the integral.

Evaluate Answer C

The snail is moving at a constant rate of 70 cm per hour for 8 hours, so again, this quantity could be represented by the integral.

Evaluate Answer D

The company is selling merchandise at a constant rate of \$70 per hour for 8 hours, so this choice could also be represented by the given integral.

Evaluate Answer E

The tide is rising at a rate of 70 mm per minute for 8 minutes, which is not the same as a rate of 70 units per hour for 8 hours. Therefore, this is the only choice that cannot be represented by the integral.

Key concepts

Constant Rate of Change

Understanding the concept of a constant rate of change is crucial when dealing with problems related to motion, economics, and various other fields. In essence, a constant rate of change means that the amount of change over each unit of time is the same. For example, if a train is moving at a rate of 70 miles per hour, it implies that for every hour of its journey, it consistently travels 70 miles.

When a quantity changes at a constant rate, we can predict its behavior over time, making calculations and projections simpler. In the scenario of the train, if we know it has been travelling for 8 hours at this unvarying speed, we can confidently state that the train will have covered a total distance of 560 miles, without any complicated calculations needed. The definite integral \( \int_{0}^{8} 70 \, dt \) portrays this type of scenario where the '70' represents the constant rate, 'dt' indicates a tiny increment of time, and 0 to 8 denotes the time period over which the rate is applied.

Integral Representation

Integral representation is a powerful mathematical concept used to solve problems involving accumulation or the area under a curve. In the context of our original exercise, the definite integral \( \int_{0}^{8} 70 \, dt \) can be perceived as the accumulation of a quantity that changes at a constant rate over a specified period.

The '70' in the integral denotes the constant rate of change, and \(dt\) signifies an infinitesimally small interval of time. The integration process essentially sums up these small contributions over the interval from 0 to 8 to provide the total quantity accumulated over the time period considered. In physics, it could represent distance traveled over time at a constant speed; in economics, it might symbolize the total sales over time at a stable rate. The integral thus materializes as a versatile tool for translating situations with a constant rate of change into precise numerical values.

Units Conversion

Units conversion is a fundamental aspect to consider when interpreting definite integrals, particularly in applied contexts where different measurement systems are used. In the context of the original problem, the integral given uses units of hours for time. It's critical to ensure that the rates are also expressed in corresponding units to result in a coherent calculation.

For instance, if a rate is given in miles per hour (mph) and time is in hours, no conversion is necessary. However, if the rate is in a different unit, such as millimeters per minute, we need to convert one of the units so that both rate and time are in compatible units. In the case provided, 70 mm per minute cannot be directly integrated over a period described in hours without first converting the minutes to hours or vice versa. Failing to do so can lead to incorrect answers and misinterpretations. Understanding how to convert units is essential in ensuring that the integration is not only mathematically correct but also contextually accurate.