Problem 2 - Applied Calculus

Question

Each of the following functions gives the amount of a substance present at time \(t\). In each case, give the amount present initially (at \(t=0\) ), state whether the function represents exponential growth or decay, and give the percent growth or decay rate. (a) \(\quad A=100(1.07)^{t}\) (b) \(A=5.3(1.054)^{t}\) (c) \(A=3500(0.93)^{t}\) (d) \(\quad A=12(0.88)^{t}\)

Step-by-Step Solution

Answer

Initial amounts: (a) 100, (b) 5.3, (c) 3500, (d) 12. Growth rates: (a) 7%, (b) 5.4%. Decay rates: (c) 7%, (d) 12%.

Step 1: Identify the Initial Amount (t=0)

For each function, substitute \( t = 0 \) into the equation for \( A \). The initial amount is the coefficient of the exponential expression, as \( b^0 = 1 \).

Step 2: Determine Growth or Decay

Look at the base of the exponent (the number inside the parentheses). If it is greater than 1, the function represents exponential growth. If it is less than 1, it represents exponential decay.

Step 3: Calculate the Percent Growth or Decay Rate

Determine the growth or decay rate by subtracting 1 from the base of the exponent (\(b = 1 + r\) where \(r\) is the rate). Multiply the result by 100 to get the percentage.

Key Concepts

Exponential Growth

Exponential growth describes a process where the quantity increases by a consistent percentage over equal time intervals. This is evident in functions where the base of the exponential term—often inside the parentheses—is greater than 1. To determine if a function exhibits exponential growth, we look at the base of the exponential expression, also known as the growth factor.
In the function, \( A = 100(1.07)^t \), the base of the exponent is 1.07. Here, 1.07 indicates that each interval of time results in a 7% increase in the initial amount. The pattern of increasing values is typical of exponential growth, which can model scenarios such as population growth or compound interest.
For solving problems involving exponential growth, follow these simple steps:

• Identify the base of the exponent. If it is greater than 1, the function represents growth.
• Subtract 1 from the base to find the growth rate \((r)\).
• Multiply \(r\) by 100 to obtain the percentage growth rate.

This approach makes it simple to quickly ascertain the nature and rate of growth in exponential functions.

Exponential Decay

Exponential decay describes the process where a quantity decreases by a consistent percentage over equal time intervals. This phenomenon is observed when the base of the exponential expression is between 0 and 1, indicating that each time interval sees a reduction in the initial amount. This type of decay is evident in processes such as radioactive decay or depreciation of an asset.
For example, in the expression \( A = 3500(0.93)^t \), the base of 0.93 signifies a 7% loss over each time period. In exponential decay scenarios,

• Find the base of the exponent. A base less than 1 indicates decay.
• Calculate the decay rate by subtracting the base from 1.
• Convert the rate into a percentage by multiplying by 100.

Understanding exponential decay is crucial for interpreting situations where things diminish at a certain rate, such as in the context of population decline or half-life of substances.

Initial Amount

The initial amount in exponential functions is the starting value when time \((t)\) equals zero. It is represented as the coefficient of the exponential term and is essential for setting benchmarks or initial conditions when analyzing growth or decay functions.
To determine the initial amount in an expression like \( A = 12(0.88)^t \), substitute \( t = 0 \), which simplifies to \( A = 12 \times 1 = 12 \). This is because raising any number to the 0 power equals 1. Therefore, in any expression \( A = P(b)^t \), \( P \) is the initial amount.

• Set \( t = 0 \) in the function and simplify.
• Recognize that the coefficient of the term \( b^t \) provides the initial amount.

Recognizing the initial amount is fundamental when graphing the function or comparing models, as it serves as the baseline from which all calculations and projections originate.