Problem 21 - Essential Calculus

Question

Describe what the values of \(C\) and \(k\) represent in the exponential growth and decay model, \(y=C e^{k t}\).

Step-by-Step Solution

Answer

In the exponential growth and decay model, \(C\) represents the initial value of the quantity being considered, and \(k\) is the growth or decay constant that indicates the rate of increase or decrease.

Step 1: Context and Understanding

First, it's crucial to understand what the exponential growth and decay model is. It's a type of mathematical model that describes a quantity that increases or decreases proportionally to the amount currently present. This model is commonly used to represent real-life growth or decay situations such as population growth, radioactive decay, or interest accumulation.

Step 2: Define 'C' in Exponential Growth/Decay Model

`C` is the initial value of the quantity at the beginning of the observation (when \(t = 0\)). It describes the quantity's size at the start of the growth or decay process. Mathematically, when \(t = 0\), the equation becomes \(y=C\), showing that \(C\) is the value of \(y\) at \(t=0\). In practical applications, this could be the initial population size, the initial amount of a radioactive substance, or the principal amount in a bank account.

Step 3: Define 'k' in Exponential Growth/Decay Model

`k` is the growth or decay constant, directly related to the rate of growth or decay. If \(k > 0\), the function represents exponential growth, and if \(k < 0\), it represents exponential decay. It determines how quickly or slowly the quantity grows or decays over time. This could represent the birth-death rate of a population, the decay constant of a radioactive substance, or the interest rate of a bank account.