exponential growth and decay

Front

What is exponential growth?

Back

Exponential growth occurs when the increase in a quantity is proportional to its current value, leading to growth at an increasing rate. For example, if a population grows by 10% each year, it grows faster each year as the base number increases.

Front

What is exponential decay?

Back

Exponential decay is the decrease in a quantity at a rate proportional to its current value, leading to a rapid decline. For example, radioactive substances decay at a rate proportional to their current amount.

Front

What is the formula for exponential growth?

Back

The formula for exponential growth is A = P(1 + r)^t, where A is the amount after time t, P is the initial amount, r is the growth rate, and t is time.

Front

What is the formula for exponential decay?

Back

The formula for exponential decay is A = P(1 - r)^t, where A is the amount after time t, P is the initial amount, r is the decay rate, and t is time.

Front

How do you find the decay rate from a decay model?

Back

In a decay model of the form A = P(1 - r)^t, the decay rate r can be found by identifying the coefficient of the exponential term.

Front

If a population of 1000 decreases by 20% each year, what is the population after 3 years?

Back

Using the formula A = P(1 - r)^t, A = 1000(1 - 0.20)^3 = 1000(0.8)^3 = 512.

Front

What does a decay factor of 0.85 indicate?

Back

A decay factor of 0.85 indicates that the quantity retains 85% of its value after each time period, meaning it decreases by 15%.

Front

What is the relationship between doubling time and exponential growth?

Back

In exponential growth, the doubling time is the period it takes for a quantity to double in size. It is constant and can be calculated using the rule of 70: Doubling time (in years) = 70 / growth rate (in %).

Front

How do you model a situation where a quantity doubles every fixed interval?

Back

To model a situation where a quantity doubles every fixed interval, use the formula A = P(2)^(t/T), where T is the time interval for doubling.

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What is the significance of the base in an exponential function?

Back

The base in an exponential function determines the rate of growth or decay. A base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.