Problem 50 - Calculus

Question

Give an example of: A formula representing the statement " \(q\) decreases at a constant percent rate, and \(q=2.2\) when \(t=0 . "\)

Step-by-Step Solution

Answer

\(q(t) = 2.2 (1 - r)^t\), where \(r\) is the decay rate.

Step 1: Understand the Exponential Decay Model

The statement "\(q\) decreases at a constant percent rate" indicates an exponential decay model. Exponential decay can be represented by the formula \(q(t) = q_0 (1 - r)^t\), where \(q_0\) is the initial quantity (when \(t = 0\)), \(r\) is the decay rate, and \(t\) is time.

Step 2: Identify Initial Condition

The problem states that \(q = 2.2\) when \(t = 0\). This means that the initial quantity \(q_0 = 2.2\). Substituting this into the general exponential decay model gives \(q(t) = 2.2 (1 - r)^t\).

Step 3: Represent the Given Information with a Formula

Since we know the initial quantity and recognize the need for an exponential decay formula, we can represent the statement as \(q(t) = 2.2 (1 - r)^t\), with \(r\) representing the constant percent rate at which \(q\) decreases.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that model situations where something changes at a constant rate over time. They are important in both growth and decay scenarios.
For exponential decay, the general form of the function is:

• \( q(t) = q_0 (1 - r)^t \)

Here:

• \( q(t) \) represents the quantity at time \( t \).
• \( q_0 \) is the initial quantity when time \( t = 0 \).
• \( r \) is the decay rate expressed as a decimal.
• \( t \) is the time period over which the quantity changes.

In exponential decay, the quantity decreases over time as a certain percentage is subtracted in each time period. This results in a curve that gradually approaches zero, never quite reaching it.

Initial Value

The initial value in an exponential function is a crucial piece of information. It refers to the starting point of the quantity being measured, which means the value of the function at time \( t = 0 \).
In our context, this is denoted by \( q_0 \). For instance, if a problem states that \( q = 2.2 \) initially, then \( q_0 = 2.2 \).
Understanding the initial value is essential because it provides the baseline for calculating future values. It’s also an input in the formula for exponential decay. Without knowing \( q_0 \), it would be impossible to accurately describe how the quantity evolves over time.
So, always ensure to pinpoint the initial value when working with exponential functions.

Decay Rate

The decay rate is the percentage at which a quantity decreases over a specific time period in an exponential decay scenario.
It is symbolized by \( r \) in the exponential decay formula.

• It is crucial to express the decay rate as a decimal. For example, a 5% decay rate would be written as \( r = 0.05 \).
• The decay rate determines how quickly the quantity shrinks. A higher decay rate means faster reduction.

In the exponential decay formula \( q(t) = q_0 (1 - r)^t \), the term \( (1 - r) \) shows that a portion of the initial value is reduced with every passing time interval.
Understanding the decay rate helps in predicting future values of the quantity, seeing how drastically it will shrink as time progresses.