Problem 13 - Calculus

Question

Let \(f(t)=Q_{0} a^{t}=Q_{0}(1+r)^{t}\) (a) Find the base, \(a\) (b) Find the percentage growth rate, \(r\) $$f(5)=75.94 \text { and } f(7)=170.86$$

Step-by-Step Solution

Answer

(a) Base: 1.5 (b) Percentage growth rate: 50%

Step 1: Setting up the equation for the base 'a'

We are given the exponential function \(f(t) = Q_0 a^t = Q_0 (1+r)^t\). To find the base \(a\), we need to use the given function values \(f(5) = 75.94\) and \(f(7) = 170.86\). Assuming \(f(t) = Q_0 a^t\), we have two equations: \(Q_0 a^5 = 75.94\) and \(Q_0 a^7 = 170.86\). We can divide these two equations to eliminate \(Q_0\) and find \(a\).

Step 2: Solving for 'a'

Divide the second equation by the first:\[ \frac{Q_0 a^7}{Q_0 a^5} = \frac{170.86}{75.94} \]This simplifies to:\[ a^2 = \frac{170.86}{75.94} \]Solving for \(a\), take the square root on both sides:\[ a = \sqrt{\frac{170.86}{75.94}} \]

Step 3: Substituting values to find 'a'

Calculate the value of \(a\) using the expression from Step 2:\[ a = \sqrt{\frac{170.86}{75.94}} \approx \sqrt{2.25} \approx 1.5 \]

Step 4: Calculating the percentage growth rate 'r'

With \(a\) found, use the relation \(a = 1 + r\) to find the percentage growth rate \(r\). Since \(a = 1.5\), we have:\[ 1 + r = 1.5 \]Thus, \(r = 0.5\) or as a percentage, \(r = 50\%\).

Key Concepts

Exponential Growth

Exponential growth refers to a process where a quantity increases by a consistent percentage over equal time periods. This is modeled by exponential functions, such as the one given in the exercise: \( f(t) = Q_0 a^t \). Here, \( Q_0 \) represents the initial quantity, \( a \) is the base of the exponential function, and \( t \) is the time period.

In real-world scenarios, exponential growth is commonly observed in populations, investments, and even technology adoption. It's important to recognize the exponential nature because it signifies rapid increases. For instance, with exponential growth, even a small initial quantity can grow to a massive number over time if the growth rate is high.

Characteristics of exponential growth include:

• Growth by a specific ratio (or percentage) over regular intervals.
• Non-linear increase, meaning it accelerates over time rather than having a steady linear path.
• The time it takes for the quantity to double (often called the doubling time) is constant if the growth rate remains unchanged.

Base of Exponential Function

The base of an exponential function, denoted as \( a \) in \( f(t) = Q_0 a^t \), is a crucial part of understanding how exponential growth operates. The base determines how quickly the growth occurs. In the context of the exercise, the base \( a \) helps quantify the growth factor.

In this exercise, we find \( a \) by using given function values at different times. By setting up equations for \( f(t) = Q_0 a^t \), and solving for \( a \), we were able to compute the base as approximately 1.5. This indicates that for every time unit, the quantity increases by a factor of 1.5.

Highlights of the Base in Exponential Functions:

• If \( a > 1 \), it represents growth since the quantity increases each period.
• If \( a = 1 \), there is no growth as the initial value remains constant over time.
• If \( a < 1 \), it would indicate exponential decay, where the quantity decreases over time.

Percentage Growth Rate

The percentage growth rate is a way to express how quickly a quantity is increasing relative to its current value. It's derived from the base \( a \) using the relation \( a = 1 + r \). Here, \( r \) is the percentage growth rate, often expressed as a decimal.

In our example, where \( a = 1.5 \), we determine the growth rate by solving \( 1 + r = 1.5 \), which gives us \( r = 0.5 \). Converted into a percentage, this growth rate is 50%. This means that each successive time period sees a 50% increase of the quantity from its previous period.

Key Points on Understanding Percentage Growth Rate:

• Crucial for predicting future values and comparing different growth scenarios.
• Simple to calculate once the base is known: subtract 1 from the base and multiply by 100 to get the percentage.
• Helps to easily communicate the magnitude of growth across different contexts and fields.