Applications of Exponentials Revision Notes for Grade 11 NSC Matric Mathematics

Applications of Exponentials

Exponential functions appear frequently in real-world situations, making them essential for solving practical problems. They are commonly used to model population growth in biology and to calculate various types of interest in finance.

Understanding exponential growth

When quantities grow exponentially, they increase by the same percentage over equal time intervals. This creates a pattern where the growth rate remains constant, but the actual amount of increase gets larger over time.

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The key characteristic of exponential growth is that the percentage increase remains constant, even though the absolute amount of increase becomes larger with each time period.

The exponential growth formula

The fundamental formula for exponential growth situations is:

$\text{Final amount} = \text{Initial amount} \times (1 + \text{growth rate})^{\text{time periods}}$

Where:

Growth rate is expressed as a decimal (e.g., 80% = 0.8)
Time periods must match the units of the growth rate

Worked example 1: Bacterial population growth

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Worked Example: Bacterial Population Growth

Problem: A bacterial colony grows at an exponential rate of 80% every hour. Starting with 10 bacteria, calculate the population after 5 hours, 1 day, and 1 week.

Solution:

Step 1: Set up the exponential formula

Since the bacteria grow at 80% per hour, the growth factor is (1 + 0.8) = 1.8

$\text{Final population} = 10 \times (1.8)^n$

where $n$ = number of hours

Step 2: Calculate population after 5 hours

$\text{Final population} = 10 \times (1.8)^5 \approx 189 \text{ bacteria}$

Step 3: Calculate population after 1 day (24 hours)

$\text{Final population} = 10 \times (1.8)^{24} \approx 13,382,588 \text{ bacteria}$

Step 4: Calculate population after 1 week (168 hours)

$\text{Final population} = 10 \times (1.8)^{168} \approx 7.687 \times 10^{43} \text{ bacteria}$

Notice how the population grows dramatically over longer time periods. Very large numbers like this are expressed in scientific notation for practical purposes.

Worked example 2: Fish population growth

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Worked Example: Fish Population Growth

Problem: A rare fish species has 821 individuals initially and grows at 2% per month. Calculate the population after 6 months, 10 years, and 100 years.

Solution:

Step 1: Set up the exponential formula

With 2% monthly growth, the growth factor is (1 + 0.02) = 1.02

$\text{Final population} = 821 \times (1.02)^n$

where $n$ = number of months

Step 2: Calculate population after 6 months

$\text{Final population} = 821 \times (1.02)^6 \approx 925 \text{ fish}$

Step 3: Calculate population after 10 years (120 months)

$\text{Final population} = 821 \times (1.02)^{120} \approx 8,838 \text{ fish}$

Step 4: Calculate population after 100 years (1200 months)

$\text{Final population} = 821 \times (1.02)^{1200} \approx 1.716 \times 10^{13} \text{ fish}$

Again, this extremely large number requires scientific notation. Even with a relatively small growth rate of 2% per month, the population becomes enormous over long time periods.

Key exam tips

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Critical Exam Guidelines:

Always check that your time units are consistent with the growth rate period
Convert time carefully (1 day = 24 hours, 1 year = 12 months, etc.)
Use scientific notation for very large numbers in your final answers
Show each calculation step clearly for full marks
Remember that the growth factor is (1 + growth rate), not just the growth rate

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Key Points to Remember:

Exponential growth formula: Final = Initial × (1 + rate)^time
Time units must match the growth rate period (hourly, monthly, yearly)
Large results should be expressed in scientific notation
Growth factor is always (1 + growth percentage as decimal)
Step-by-step working is essential for exam success

Applications of Exponentials (Grade 11 NSC Matric Mathematics): Revision Notes