Exponential growth and decay Revision Notes

Exponential growth and decay describe processes where a quantity changes by a consistent percentage over equal intervals of time. This means the amount increases or decreases proportionally to its current value, not by a fixed amount.

In exponential growth, the quantity becomes larger each period because it grows by a fixed percentage of its current size. For example, a population increasing by 5% every year grows faster as it gets bigger.

In exponential decay, the quantity reduces by a fixed percentage each period. For example, the value of a car depreciating by 10% annually decreases more in the first years than later, but always by 10% of its current value.

• Growth means increasing by a fixed percentage per period.
• Decay means decreasing by a fixed percentage per period.
• The change depends on the current amount, not a fixed amount.

We use formulas to calculate the amount after a number of time periods when growth or decay occurs.

For exponential growth, the formula is: \( A = P(1 + r)^t \) where:

\( A \) is the amount after \( t \) periods, \( P \) is the initial amount, \( r \) is the growth rate per period expressed as a decimal, \( t \) is the number of time periods (e.g., years). For exponential decay, the formula is similar but subtracts the rate: \( A = P(1 - r)^t \). Here, \( r \) is the decay rate as a decimal. These formulas allow precise calculation of future or reduced amounts without needing the constant e.

• Growth formula: \( A = P(1 + r)^t \)
• Decay formula: \( A = P(1 - r)^t \)
• \( P \) = initial amount, \( r \) = rate as decimal, \( t \) = time periods

Suppose a town has a population of 12,000 people and it grows by 3% each year. Find the population after 5 years.

Using the growth formula: \( A = P(1 + r)^t \) Substitute: \( P = 12,000 \), \( r = 0.03 \), \( t = 5 \) Calculate: \( A = 12,000 \times (1 + 0.03)^5 = 12,000 \times (1.03)^5 \) Calculate \( (1.03)^5 \): approximately 1.159274 Multiply: \( 12,000 \times 1.159274 = 13,911.29 \) Rounded to the nearest whole number, the population after 5 years is approximately 13,911 people.

This shows how the population increases more each year because the growth is on the current population, not the original 12,000.

• Identify initial amount, rate, and time.
• Use growth formula with decimal rate.
• Calculate power and multiply.
• Round final answer appropriately.

A car is worth £20,000 when new and depreciates by 15% each year. Find its value after 3 years.

Using the decay formula: \( A = P(1 - r)^t \) Substitute: \( P = 20,000 \), \( r = 0.15 \), \( t = 3 \) Calculate: \( A = 20,000 \times (1 - 0.15)^3 = 20,000 \times (0.85)^3 \) Calculate \( (0.85)^3 \): approximately 0.614125 Multiply: \( 20,000 \times 0.614125 = 12,282.50 \) The car’s value after 3 years is approximately £12,283.

This demonstrates how the value decreases by 15% of the current value each year, not a fixed £3,000 per year.

• Identify initial value, decay rate, and time.
• Use decay formula with decimal rate.
• Calculate power and multiply.
• Round final answer appropriately.

Exponential growth and decay models apply to many real-life situations:

Population growth where birth rates and migration cause increases proportional to current population.

Depreciation of assets like vehicles, machinery, or electronics losing value over time at a fixed percentage rate per year or month. Radioactive decay (though this often involves the constant e, which is beyond this level). Understanding these models helps predict future values or sizes and make informed decisions, such as estimating future population or asset worth.

• Population changes often follow exponential growth.
• Asset depreciation follows exponential decay.
• Models help in planning and forecasting.

Always convert percentage rates to decimals before substituting into formulas.

The time period \( t \) must match the rate period (e.g., years if rate is per year).

The constant e is not required for calculations at this level; use the (1 ± r)^t formulas instead. Check units and round answers appropriately for the context. Understand the difference between growth (adding) and decay (subtracting) in the formulas.

• Convert percentages to decimals.
• Match time units with rate periods.
• Use (1 ± r)^t formulas, not e.
• Check units and round carefully.
• Distinguish growth from decay formulas.