Problem 43 - Calculus

Question

Let \(W\) be the amount of water, in gallons, in a bathtub at time \(t,\) in minutes. (a) What are the meaning and units of \(d W / d t ?\) (b) Suppose the bathtub is full of water at time \(t_{0},\) so that \(W\left(t_{0}\right)>0 .\) Subsequently, at time \(t_{p}>\) \(t_{0}.\) the plug is pulled. Is \(d W / d t\) positive, negative, or zero: (i) For \(t_{0}

Step-by-Step Solution

Answer

(a) Rate of water change in gallons/minute. (i) Zero, (ii) Negative, (iii) Zero.

Step 1: Understanding the Derivative

The notation \(dW/dt\) represents the rate of change of the amount of water \(W\) in the tub over time \(t\). The units for \(dW/dt\) are gallons per minute \((\text{gallons/minute})\), as it describes how many gallons are being added or removed per minute from the tub.

Step 2: Evaluating the Scenario Before the Plug is Pulled

For the time interval before the plug is pulled, i.e., \(t_0 < t < t_p\), the rate at which water changes in the tub is zero if no water is being added or removed. Therefore, \(dW/dt = 0\).

Step 3: Evaluating the Scenario After the Plug is Pulled

Once the plug is pulled at time \(t_p\), the water starts draining out of the tub. Hence, for \(t > t_p\) but before the tub is fully empty, \(dW/dt\) becomes negative since the amount of water is decreasing.

Step 4: Evaluating When the Tub is Empty

When all the water has drained out, \(W = 0\). At this point, since there is no more water to drain, \(dW/dt = 0\).

Key Concepts

Derivatives

Derivatives in calculus are a powerful tool used to analyze how a function changes at any given point. They are essentially the mathematical way of expressing the concept of change or slope. A derivative provides us the rate at which one quantity changes with respect to another.
For example, in our bathtub problem, the derivative \(\frac{dW}{dt}\) represents how the water level in the tub changes over time. This tells us how fast or slow the water is either filling up or draining out of the tub at any moment.

• Expressing derivatives as \(\frac{dW}{dt}\) shows the instantaneous rate of change of \(W\) concerning \(t\).
• The derivative can be visually imagined as the slope of the tangent line to the curve of the function at a specific point.

This concept of the derivative allows us to predict behavior and make calculations regarding the dynamics of real-world functions.

Rate of Change

Rate of change is a key concept closely tied to derivatives. It tells us how a quantity varies with time or with another variable.
In the bathtub example, the rate of change is represented by \(\frac{dW}{dt}\), indicating how the volume of water changes over time.

• If \(\frac{dW}{dt} > 0\), the water level is increasing over time, which means water is being added to the tub.
• If \(\frac{dW}{dt} < 0\), the water level is decreasing, indicating that water is draining out.
• If \(\frac{dW}{dt} = 0\), the water level remains constant, meaning no water is being added or removed.

By understanding the rate of change, we gain insight into how systems behave over time which is crucial for predicting future conditions or understanding past behaviors.

Units of Measurement

In calculus and other fields, units of measurement are essential for interpreting and understanding results. They are particularly important when dealing with derivatives because they provide context for what the derivative signifies.
In the given problem, the derivative \(\frac{dW}{dt}\) is measured in gallons per minute.

• Gallons, a unit of volume, is used here to describe the quantity of water in the bathtub.
• Minutes, a unit of time, gives us the duration over which the rate of change is measured.

Together, these units help us understand that \(\frac{dW}{dt}\) specifies how rapidly the water volume changes over time, usually in practical, real-world scenarios like filling or draining a bathtub.

Real-World Applications

The concept of derivatives extends far beyond theoretical mathematics and finds utility in numerous real-world applications. They are essential for modeling and solving issues in various fields like physics, engineering, economics, and biological sciences.
In our bathtub scenario, understanding the derivative \(\frac{dW}{dt}\) allows us to model how quickly the bathtub fills up or drains, which can be useful in designing efficient plumbing systems.
Other applications of derivatives in the real world include:

• Designing car engines that can change speed (acceleration and deceleration).
• Predicting future trends in economics such as investment growth rates.
• Analyzing the spread of diseases by looking at the rate of infection in epidemiology.

Understanding these applications of derivatives shows their vital role in helping us comprehend and manipulate the world around us, making them a powerful tool for both engineers and scientists alike.