Worksheet on exponential growth functions, including definitions, equations, and practice problems.

Problem Analysis and Solution

The provided worksheet focuses on exponential growth and its application in financial contexts, specifically compound interest. Let's solve each question step by step.

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#### 1) What is exponential growth?
Exponential growth refers to a pattern of increasing values where the rate of growth is proportional to the current value. In other words, the quantity grows by a fixed percentage over equal intervals of time. Mathematically, it is represented by the formula:
\[
A = C \cdot (1 + r)^t
\]
where:
- \( A \) is the amount after time \( t \),
- \( C \) is the initial amount,
- \( r \) is the growth rate (as a decimal),
- \( t \) is the time elapsed.

Key characteristics of exponential growth include:
- The growth rate is constant.
- The growth becomes increasingly rapid over time.

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#### 2) Write the growth model in terms of \( C \), \( r \), and \( t \).
The general formula for exponential growth is:
\[
A = C \cdot (1 + r)^t
\]
Here:
- \( C \) is the initial amount,
- \( r \) is the growth rate (as a decimal),
- \( t \) is the time elapsed,
- \( 1 + r \) is the growth factor.

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#### 3) What is the initial amount \( P \)?
The problem states that you deposited \$10,000 in a CD account. This deposit represents the initial amount before any growth occurs. Therefore:
\[
P = 10,000
\]

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#### 4) What is the growth rate \( r \)?
The problem specifies that the CD account pays an interest rate of 11% compounded yearly. Since the growth rate \( r \) must be expressed as a decimal, we convert 11% to a decimal:
\[
r = \frac{11}{100} = 0.11
\]

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#### 5) Complete the equation: \( A = \_\_\_ (1 + \_\_\_)^t \) after \( t \) years.
Using the general formula for exponential growth:
\[
A = C \cdot (1 + r)^t
\]
From the previous questions:
- \( C = 10,000 \)
- \( r = 0.11 \)

Substitute these values into the formula:
\[
A = 10,000 \cdot (1 + 0.11)^t
\]

Thus, the completed equation is:
\[
A = 10,000 \cdot (1.11)^t
\]

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#### 6) Find your balance after 5 years using the equation from question number 5.
We use the equation derived in question 5:
\[
A = 10,000 \cdot (1.11)^t
\]
Here, \( t = 5 \). Substitute \( t = 5 \) into the equation:
\[
A = 10,000 \cdot (1.11)^5
\]

First, calculate \( (1.11)^5 \):
\[
(1.11)^5 \approx 1.68506
\]

Now, multiply by the initial amount:
\[
A = 10,000 \cdot 1.68506 \approx 16,850.60
\]

Thus, the balance after 5 years is:
\[
A \approx 16,850.60
\]

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#### 7) Identify the initial amount and growth factor in the following exponential functions:
We are given three exponential functions, and we need to identify the initial amount (\( C \)) and the growth factor (\( 1 + r \)) for each.

##### Function 1: \( y = 300(1 + 0.2)^t \)
- Initial amount (\( C \)): 300
- Growth factor: \( 1 + 0.2 = 1.2 \)

##### Function 2: \( y = 220(1 + 8)^t \)
- Initial amount (\( C \)): 220
- Growth factor: \( 1 + 8 = 9 \)

##### Function 3: \( y = 400(1.8)^t \)
- Initial amount (\( C \)): 400
- Growth factor: 1.8 (already in the form \( 1 + r \))

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Final Answers:

1. Exponential growth is a pattern where the rate of increase is proportional to the current value.
2. The growth model is \( A = C \cdot (1 + r)^t \).
3. The initial amount \( P \) is \( \boxed{10,000} \).
4. The growth rate \( r \) is \( \boxed{0.11} \).
5. The completed equation is \( A = 10,000 \cdot (1.11)^t \).
6. The balance after 5 years is \( \boxed{16,850.60} \).
7.
- For \( y = 300(1 + 0.2)^t \): Initial amount = 300, Growth factor = 1.2.
- For \( y = 220(1 + 8)^t \): Initial amount = 220, Growth factor = 9.
- For \( y = 400(1.8)^t \): Initial amount = 400, Growth factor = 1.8.