Exponential Growth and Decay Review

Front

What is exponential growth?

Back

Exponential growth occurs when a quantity increases by a constant percentage over a period of time, resulting in a rapid increase. For example, if a population grows by 10% each year, it will grow faster each subsequent year.

Front

What is exponential decay?

Back

Exponential decay occurs when a quantity decreases by a constant percentage over time, leading to a rapid decrease. For example, if a population decreases by 5% each year, it will decline faster each subsequent year.

Front

What is the general formula for exponential growth?

Back

The general formula for exponential growth is y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Front

What is the general formula for exponential decay?

Back

The general formula for exponential decay is y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

Front

How do you determine if a function represents exponential growth or decay?

Back

If the base of the exponential function is greater than 1, it represents growth. If the base is between 0 and 1, it represents decay.

Front

What is the effect of a 10% decay rate on a population of 1000 after 1 year?

Back

After 1 year, the population would be 1000(1 - 0.10) = 900.

Front

If a population of 500 decreases by 4% each year, what will be the population after 3 years?

Back

Population after 3 years = 500(1 - 0.04)^3 ≈ 500(0.864) ≈ 432.

Front

What does the term 'half-life' refer to in exponential decay?

Back

Half-life is the time required for a quantity to reduce to half its initial value. It is commonly used in contexts like radioactive decay.

Front

If a quantity doubles every 5 years, what is its growth rate?

Back

The growth rate can be calculated using the formula r = (2^(1/t) - 1), where t is the doubling time. For 5 years, r ≈ 0.1487 or 14.87%.

Front

What is the population of a city if it is currently 200,000 and is expected to grow at a rate of 3% per year for 10 years?

Back

Population after 10 years = 200,000(1 + 0.03)^10 ≈ 200,000(1.3439) ≈ 268,780.