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) \(\quad A=5.3(1.054)^{t}\) (c) \(\quad A=3500(0.93)^{t}\) (d) \(\quad A=12(0.88)^{t}\)

Step-by-Step Solution

Answer

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

Step 1: Find Initial Amount for (a)

For function \( A = 100(1.07)^t \), the initial amount is the value of \( A \) when \( t = 0 \). Substituting \( t = 0 \), we get:\[A = 100(1.07)^0 = 100 \]Thus, the initial amount is 100.

Step 2: Determine Growth or Decay for (a)

The expression inside the parentheses for this function is \(1.07\), which is greater than 1. This represents exponential growth.

Step 3: Calculate Percent Growth for (a)

The growth factor is \(1.07\). To find the percent growth rate, subtract 1 and multiply by 100:\[(1.07 - 1) \times 100 = 7\%\]So, the percent growth rate is 7%.

Step 4: Find Initial Amount for (b)

For function \( A = 5.3(1.054)^t \), the initial amount at \( t = 0 \) is calculated as:\[A = 5.3(1.054)^0 = 5.3\]Thus, the initial amount is 5.3.

Step 5: Determine Growth or Decay for (b)

The base \(1.054\) is greater than 1, indicating exponential growth.

Step 6: Calculate Percent Growth for (b)

The growth factor is \(1.054\). To find the percent growth rate:\[(1.054 - 1) \times 100 = 5.4\%\]So, the percent growth rate is 5.4%.

Step 7: Find Initial Amount for (c)

For function \( A = 3500(0.93)^t \), calculate the initial amount at \( t = 0 \):\[A = 3500(0.93)^0 = 3500\]Thus, the initial amount is 3500.

Step 8: Determine Growth or Decay for (c)

The base \(0.93\) is less than 1, indicating exponential decay.

Step 9: Calculate Percent Decay for (c)

The decay factor is \(0.93\). To find the percent decay rate:\[(1 - 0.93) \times 100 = 7\%\]So, the percent decay rate is 7%.

Step 10: Find Initial Amount for (d)

For function \( A = 12(0.88)^t \), the initial amount at \( t = 0 \) is:\[A = 12(0.88)^0 = 12\]Thus, the initial amount is 12.

Step 11: Determine Growth or Decay for (d)

The base \(0.88\) is less than 1, indicating exponential decay.

Step 12: Calculate Percent Decay for (d)

The decay factor is \(0.88\). To find the percent decay rate:\[(1 - 0.88) \times 100 = 12\%\]So, the percent decay rate is 12%.

Key Concepts

Exponential Growth

Exponential growth occurs when the amount of a substance increases over time. This pattern can be seen in natural phenomena such as populations or investments.

When you encounter an exponential function like \(A = 100(1.07)^t\), the number 1.07 within the parentheses is crucial. If this number, also known as the growth factor, is greater than 1, it indicates exponential growth.

• Easy to spot: If the growth factor is more than 1.
• The effect: The quantity gets larger as time progresses.
• Example: A growing population or increasing interest on savings.

In our example for functions \(A = 100(1.07)^t\) and \(A = 5.3(1.054)^t\), the substances are experiencing exponential growth because 1.07 and 1.054 are greater than 1. This tells us that at each time period \(t\), the amount increases based on that growth factor.

Exponential Decay

Exponential decay is the process by which a quantity steadily decreases over time. This concept is often applied to things like radioactive decay or depreciation of assets.

In an exponential function, such as \(A = 3500(0.93)^t\) or \(A = 12(0.88)^t\), if the number inside the parentheses is less than 1, it signals decay.

• Easy to identify: If the decay factor is less than 1.
• The impact: The amount of substance declines as time moves on.
• Example: A melting ice cube or a deflating balloon.

For these functions, with the decay factors of 0.93 and 0.88 respectively, we see that they represent exponential decay. Each time period brings a further reduction in the quantity by a consistent factor.

Percent Growth Rate

The percent growth rate shows how much a quantity grows per time period in percentage terms. This number gives a clearer picture than the growth factor alone.

To calculate the percent growth rate from a growth factor, subtract 1 from the growth factor and then multiply by 100:

• First step: Subtract 1 from the growth factor.
• Second step: Multiply the result by 100 to convert it to a percentage.

For example, if we have a growth factor of 1.07, the percent growth rate is \[(1.07 - 1) \times 100 = 7\%\].This means every period, the amount grows by 7%.

In our examples, the functions \(A = 100(1.07)^t\) and \(A = 5.3(1.054)^t\) have percent growth rates of 7% and 5.4% respectively. This percentage reflects how much the quantity increases every time unit.

Percent Decay Rate

The percent decay rate is useful for understanding how much a quantity decreases in percentage terms over each time period.

Here's how to find the percent decay rate from a decay factor:

• First step: Subtract the decay factor from 1.
• Second step: Multiply by 100 to express it as a percentage.

For instance, with a decay factor of 0.93, the percent decay rate is \[(1 - 0.93) \times 100 = 7\%\].This value tells us the amount decreases by 7% each period.

The functions \(A = 3500(0.93)^t\) and \(A = 12(0.88)^t\) illustrate exponential decay with percent decay rates of 7% and 12%, respectively. These rates signify the percentage of the original quantity that diminishes in every time increment.