Exponential Growth Applications

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 over time.

Front

What is the formula for calculating exponential growth?

Back

The formula 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 the time period.

Front

If a population of bacteria doubles every hour, how would you express the population after 5 hours if the initial population is 100?

Back

The population after 5 hours would be expressed as 100(2^5) = 3200.

Front

What does a growth rate of 5% per year mean in terms of exponential growth?

Back

It means that each year, the value increases by 5% of its current value, leading to a compounding effect over time.

Front

How do you calculate the future value of an investment with a fixed annual growth rate?

Back

Use the formula A = P(1 + r)^t, substituting P for the initial investment, r for the growth rate, and t for the number of years.

Front

What is the difference between linear growth and exponential growth?

Back

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

Front

If a cell phone subscriber base grows from 285 to 43871 in 9 years, what is the average annual growth rate?

Back

To find the average growth rate, use the formula r = (A/P)^(1/t) - 1, where A is the final amount, P is the initial amount, and t is the time in years.

Front

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

Back

The base determines the rate of growth; a larger base results in faster growth.

Front

How can you determine if a function represents exponential growth?

Back

A function represents exponential growth if it can be expressed in the form y = ab^x, where a > 0, b > 1.

Front

What is the value of an investment of $30,000 after 51 years at a 5% annual growth rate?

Back

Using the formula A = 30000(1 + 0.05)^51, the value is approximately $344,022.