Problem 21 - Applied Calculus

Question

The thickness, \(P,\) in \(\mathrm{mm}\), of pelican eggshells depends on the concentration, \(c,\) of \(\mathrm{PCBs}\) in the eggshell, measured in ppm (parts per million); that is, \(P=f(c)\) (a) The derivative \(f^{\prime}(c)\) is negative. What does this tell you? (b) Give units and interpret \(f(200)=0.28\) and \(f^{\prime}(200)=-0.0005\) in terms of \(\mathrm{PCBs}\) and eggshells.

Step-by-Step Solution

Answer

As PCBs increase, eggshells thin; at 200 ppm, thickness is 0.28 mm, decreasing by 0.0005 mm per ppm.

Step 1: Understanding what a Negative Derivative Means

Since the derivative \( f'(c) \) is negative, this indicates that the function \( f(c) \) is decreasing as \( c \) increases. This tells us that the thickness of the pelican eggshells, \( P \), decreases as the concentration of PCBs, \( c \), in the eggshell increases. The more PCBs present, the thinner the eggshells become.

Step 2: Analyzing the Value of \( f(200) \)

The notation \( f(200) = 0.28 \) means that when the concentration of PCBs in the eggshell is 200 ppm, the thickness of the pelican eggshell is 0.28 mm. The concentration of 200 ppm corresponds directly to a specific observed thickness of the eggshell.

Step 3: Understanding the Units and Interpretation of \( f'(200) \)

The value \( f'(200) = -0.0005 \) means that for every 1 ppm increase in PCBs at a concentration of 200 ppm, the thickness of the eggshell decreases by 0.0005 mm. This provides a rate of change of thickness with respect to the increase of PCBs concentration at 200 ppm, showing a continued decrease in eggshell thickness as PCB levels rise.

Key Concepts

Decreasing function

Imagine you're monitoring how the thickness of pelican eggshells changes as the concentration of harmful substances, like PCBs, in the shell increases. When mathematicians say a function is decreasing, it simply means that as the input (in this case, the concentration of PCBs) grows, the output (the eggshell thickness) shrinks. In mathematics, this is shown by the derivative. A negative derivative tells us that the associated function is decreasing. Here,

• Negative derivative: \(f'(c)<0\)
• Means a decreasing function.
• The derivative of thickness function \(P=f(c)\), in terms of concentration \(c\), is negative.

In practical terms, the more pollutants found in the eggshell, the thinner it becomes. This finding raises concerns about the health of pelican populations where PCB contamination is common.

Rate of change

The rate of change is a concept closely tied to derivatives. By examining the rate of change, we can understand how quickly or slowly one quantity affects another. For the thickness of pelican eggshells, the rate of change is negative, specifically

• \(f'(200)=-0.0005 \ mm/ppm\)
• This implies that for every unit increase in PCBs, eggshell thickness decreases by 0.0005 mm.

Understanding this numeric value is important because it provides clear information on the sensitivity of the eggshell thickness to changes in PCB concentration. In areas where PCB pollution varies, knowing the rate at which it impacts eggshell thickness helps in crafting conservation measures to protect wildlife.

Unit interpretation

Units serve an important role in understanding physical problems. Whenever we interpret a mathematical sentence or result, connecting the numbers to actual units, gives the situation context.Let's look at

• \(f(200)=0.28\): Measured as 0.28 mm, this is the thickness of the eggshell when PCBs are at 200 ppm.
• \(f'(200)=-0.0005\): It has units of \( mm/ppm\), making it a rate of change.

By interpreting these units, we connect mathematics to reality. The first figure gives us an exact thickness measurement, while the second shows how this thickness changes. Comprehending this information provides actionable insights for addressing the threats posed by contaminants like PCBs to bird populations.