Problem 38 - Calculus and its applications

Question

It is known that \(45 \%\) of all aluminum cans distributed are recycled each year. A beverage company uses 250,000 lb of aluminum cans. After recycling, the amount of aluminum, in pounds, still in use after \(t\) years is given by \(N(t)=250,000(0.45)^{t}\). a) Find \(N^{\prime}(t)\). b) Interpret the meaning of \(N^{\prime}(t)\).

Step-by-Step Solution

Answer

a) \(N'(t) \approx -199,625 \times (0.45)^t\); b) It shows the rate at which the aluminum amount decreases annually.

Step 1: Write Down the Given Function

The amount of aluminum, in pounds, still in use after \(t\) years is given by the function \(N(t) = 250,000(0.45)^{t}\). This is a function of exponential decay.

Step 2: Differentiate the Function

To find \(N'(t)\), we need to differentiate \(N(t) = 250,000(0.45)^t\) with respect to \(t\). The derivative of \(a^t\) where \(a\) is a constant is \(a^t \ln(a)\). Thus, the derivative of \(N(t)\) is: \[ N'(t) = 250,000 \times (0.45)^t \times \ln(0.45) \] Using \(\ln(0.45) \approx -0.7985\): \[ N'(t) \approx 250,000 \times (0.45)^t \times (-0.7985) \] So, the derivative is: \[ N'(t) \approx -199,625 \times (0.45)^t \]

Step 3: Interpret the Result

\(N'(t)\) represents the rate of change of the amount of aluminum in use with respect to time. Since \(N'(t)\) is negative, it indicates the amount of aluminum in use decreases each year. The coefficient \(-199,625\) shows how fast the amount decreases for each unit of \(t\).

Key Concepts

Differentiation

Differentiation is a key concept in calculus that allows us to find the rate at which a function is changing at any given point. In simple terms, it's like finding the slope of a curve at a particular point, instead of the entire line like regular slopes.

To find the derivative of a function, we apply certain rules. For the given exercise, we are looking at the function that describes how much aluminum remains after recycling each year. This is known as an exponential decay function.

In differentiation, we applied the differentiation rule for exponential functions, which states that if you have a function of the form \(a^t\) where \(a\) is a constant:

• The derivative is \(a^t \cdot \ln(a)\).

So, for the function \(N(t) = 250,000(0.45)^t\), applying the differentiation rule gives us:

• \(N'(t) = 250,000 \cdot (0.45)^t \cdot \ln(0.45)\).

This tells us how quickly the amount of aluminum changes with respect to time.

Exponential Function

Exponential functions are a type of mathematical function where the variable appears as an exponent. They are of the form \(a^x\), where \(a\) is a positive constant known as the base and \(x\) is the variable. One common characteristic of exponential functions is their rapid growth or decay.

In the exercise, the function \(N(t) = 250,000(0.45)^t\) is an exponential decay function. This type of function describes processes where a quantity decreases steadily but not linearly over time, such as cooling of objects, radioactive decay, or in this case, recycling.
The base \(0.45\) is less than 1, which is why we see decay, meaning the amount is reducing over time.

Exponential decay functions are valuable in real-world applications because they give us insights into how processes evolve. They allow us to predict values at future points in time based on the current trend.

Rate of Change

The rate of change in mathematics provides insight into how much something changes in relation to another variable. In the context of the exercise, we are focusing on how fast the amount of aluminum still in use is decreasing over time.

When you differentiate a function like \(N(t)\), the derivative \(N'(t)\) offers a clear picture of this rate. Since \(N'(t) \approx -199,625 \times (0.45)^t\), we know the change is negative, confirming the continuous decrease with time.

• The negative sign indicates a decline.
• The coefficient \(-199,625\) helps us understand the speed of this decline each year.

This information is crucial for companies tracking recycling efficiency and planning resources. Understanding rate of change helps organizations make data-driven decisions, predicting future trends and adapting processes accordingly.