Exponential Word Problems

Front

What is an exponential growth model?

Back

An exponential growth model describes a process where the quantity increases at a rate proportional to its current value, often represented by the formula: P(t) = P_0 * e^(rt), where P_0 is the initial amount, r is the growth rate, and t is time.

Front

How do you calculate the future value of an investment with compound interest?

Back

The future value (FV) can be calculated using the formula: FV = P(1 + r/n)^(nt), where P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.

Front

What does it mean for a population to increase by a certain percentage each year?

Back

It means that each year, the population grows by that percentage of its current size. For example, a 15% increase means the population grows by 15% of its current number every year.

Front

What is the formula for exponential growth in terms of time?

Back

The formula is: N(t) = N_0 * (1 + r)^t, where N(t) is the population at time t, N_0 is the initial population, r is the growth rate (as a decimal), and t is the time in years.

Front

How do you model the growth of a quantity that increases by a fixed percentage each time period?

Back

You can model it using the formula: 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 the number of time periods.

Front

What is the significance of the base 'e' in exponential growth?

Back

The base 'e' (approximately 2.718) is used in continuous growth models, representing the natural growth rate of processes that grow continuously rather than at discrete intervals.

Front

How do you determine the growth rate from an exponential growth problem?

Back

The growth rate can be determined by rearranging the exponential growth formula to solve for r, often using logarithms to isolate r.

Front

What is the difference between linear and exponential growth?

Back

Linear growth increases by a constant amount over time, while exponential growth increases by a percentage of the current amount, leading to faster growth as the quantity increases.

Front

How can you apply exponential growth models to real-world scenarios?

Back

Exponential growth models can be applied to populations, investments, and any scenario where growth accelerates over time, such as technology adoption or viral spread.

Front

What is the formula for calculating the value of an investment after a certain number of years with compound interest?

Back

The formula is: A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years.