Exponential Word Problems

Front

What is an exponential growth function?

Back

An exponential growth function is a mathematical expression that describes a quantity increasing at a constant percentage rate over time, typically represented as f(t) = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Front

What is the formula for compound interest?

Back

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

Front

How do you calculate the future value of an investment?

Back

To calculate the future value of an investment, use the formula FV = PV(1 + r)^n, where FV is the future value, PV is the present value (initial investment), r is the annual interest rate (as a decimal), and n is the number of years.

Front

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

Back

A population decreasing by a certain percentage each year means that the population reduces by that percentage of its current size annually, which can be modeled using an exponential decay function.

Front

What is the difference between exponential growth and exponential decay?

Back

Exponential growth occurs when a quantity increases by a constant percentage over time, while exponential decay occurs when a quantity decreases by a constant percentage over time.

Front

If an investment of $1,000 grows to $1,500 in 5 years, what is the annual growth rate?

Back

To find the annual growth rate, use the formula r = (FV/PV)^(1/n) - 1. Here, FV = 1500, PV = 1000, and n = 5. The growth rate is approximately 8.45%.

Front

What is the formula for exponential decay?

Back

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

Front

How do you model a situation where a quantity increases by a percentage each year?

Back

You can model this situation using the exponential growth function f(t) = a(1 + r)^t, where 'a' is the initial quantity, 'r' is the growth rate, and 't' is the number of years.

Front

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.

Front

How do you determine the time it takes for an investment to double?

Back

To determine the time it takes for an investment to double, use the Rule of 72: divide 72 by the annual interest rate (as a percentage). For example, at a 6% interest rate, it takes approximately 12 years to double.