Problem 17 - Calculus

Question

(a) Define the variables. (b) Write a differential equation to describe the relationship. (c) Solve the differential equation. In \(2007,\) Grinnell Glacier in Glacier National Park covered 142 acres and was estimated to be shrinking exponentially at a continuous rate of \(4.3 \%\) per year. \(^{6}\).

Step-by-Step Solution

Answer

The area of the glacier at any time \( t \) is \( A(t) = 142 e^{-0.043t} \).

Step 1: Define the Variables

Let \( A(t) \) represent the area of Grinnell Glacier in acres at time \( t \) years since 2007. Given that the initial area, \( A(0) \), is 142 acres, and that the glacier is shrinking at a continuous rate of \( 4.3\% \) per year.

Step 2: Formulating the Differential Equation

The relationship describing exponential decay can be given by the differential equation: \( \frac{dA}{dt} = -rA \), where \( r \) is the rate of decay as a decimal. Here, \( r = 0.043 \), so the equation becomes \( \frac{dA}{dt} = -0.043A \).

Step 3: Solving the Differential Equation

To solve the differential equation \( \frac{dA}{dt} = -0.043A \), we separate variables: \( \frac{1}{A} dA = -0.043 dt \). Integrating both sides, we obtain \( \ln |A| = -0.043t + C \). Solving for \( A \), we get \( A = e^{C} e^{-0.043t} \). Using the initial condition \( A(0) = 142 \), we find \( C \) so that \( 142 = e^{C} \), thus \( e^C = 142 \). Therefore, the solution is \( A(t) = 142e^{-0.043t} \).

Key Concepts

Exponential Decay

Exponential decay is a type of decay where the quantity decreases at a rate proportional to its current value. This means that the larger the quantity is, the faster it will shrink over time. In the exercise, the area of Grinnell Glacier decreases continuously at a rate of 4.3% per year. The concept of exponential decay is crucial because it helps describe processes in natural and applied sciences.

In general, the exponential decay model is expressed as:

• \[ \frac{dA}{dt} = -rA \]
• Where \( A \) is the quantity changing over time, \( t \) is time, and \( r \) is the decay rate expressed as a decimal.

This formula implies that the rate of change of the area \( A \) with respect to time is proportional to the area itself, with the rate constant \( r \). When a problem involves whether something is naturally decreasing at a steady percentage rate, exponential decay is the go-to model.

Initial Value Problem

An initial value problem is a type of differential equation that requires not just solving the equation, but also satisfying a specific, initial condition. This condition is given at some point, often at the start of the observed time period.

In the context of the exercise, the initial area of Grinnell Glacier is a critical piece of information. It is specified at time \( t = 0 \), corresponding to the year 2007, when the area was \( 142 \) acres.

• Initial conditions allow us to solve for the constant \( C \) in the integration process of a differential equation.
• This helps us find a solution that not only follows the decay law but also fits the real-world scenario given by the problem.

For this problem, solving the initial value component allows us to determine the specific expression \( A(t) = 142e^{-0.043t} \) that describes the glacier's area over time.

Separation of Variables

Separation of variables is a technique used to solve certain differential equations. The essential idea is to rearrange the equation so all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the opposite side.

In the exercise, this technique was applied in the following way:

• Start with the differential equation \( \frac{dA}{dt} = -0.043A \).
• Reorganize it to separate the variables: \( \frac{1}{A} dA = -0.043 dt \).

This set up allows each side to be independently integrated:

• The left side with respect to \( A \), and the right side with respect to \( t \).

By doing so, it becomes possible to solve for \( A \) as a function of \( t \) and thus find a general solution to the differential equation. This solution is instrumental in understanding how the glacier's area changes over time.