Exponential growth is a pattern where a quantity increases over time at a constant percentage rate. For example, the number of blogs increased at a monthly rate of about 15% over one year.
The most commonly used version of the exponential formula is:
y = a(1 + r)t
where the beginning value is a, the time is t, the end value is y, and the rate of change is r in decimal form.
Key Points
- For growth, use 1+r
- For decay, use 1−r
- Convert percentage to decimal (e.g., 5% = 0.05).
Continuous Exponential Growth
Continuous exponential growth occurs when a quantity increases continuously over time, rather than at fixed intervals like yearly or monthly.
y = aert
where
- a: initial value
- r: growth rate (in decimal)
- t: time
- e: constant ≈ 2.718
Sample Problems
Problem 1. A $100 gift card is the first prize in a radio station contest. A name is announced once a day. If the person does not call within 15 minutes, the award will be increased by 2.5 percent the next day. If there are no winners after t days, write an equation to express the value of the gift card in dollars.
The equation for exponential growth is y = a(1 + r)t.
We have, a = 100, r = 2.5% or 0.025
⇒ y = 100(1 + 0.025)t
y = 100(1.025)t
In the equation y = 100(1.025)t, y is the amount of the gift card and t is the number of days since the contest began.
Problem 2. Suppose that there is no winner after 10 days in the above problem. Determine the value of the gift card.
As per the above problem, y = 100(1.025)t.
Here, t = 10. Then,
y = 100(1.025)10
y = 128.01
The value of gift card in 10 days would be $128.01.
Problem 3. Since 2000, the cost of attending college has increased by 5% each year. Write an equation for the amount of tuition, t years after 2000 if the tuition in 2000 was $10850.
The equation for exponential growth is y = a(1 + r)t.
We have, a = $10850, r = 5% or 0.05
⇒ y = 10850(1 + 0.05)t
⇒ y = 10850(1.05)t
Problem 4. What would be the tuition fee in 2015 for the above problem?
As per the above problem, y = 10850(1.025)t.
Here, t = 2015 - 2000 = 15. Then,
y = 10850(1.05)15
⇒ y = $22555
Problem 5. In 2010, a gym sold 550 memberships. Subscriptions have climbed by 3% per year since then. For t years, write an equation to reflect the number of memberships sold.
The equation for exponential growth is y = a(1 + r)t.
We have, a = 550, r = 3% or 0.03
⇒ y = 550(1 + 0.03)t
⇒ y = 550(1.03)t
In the equation y = 550(1.03)t, y is the number of subscriptions sold and t is the number of years.
Problem 6. Find the number of memberships sold by the gym in 2020 in the above formula.
As per the above problem, y = 550(1.03)t.
Here, t = 2020 - 2010 = 10. Then,
y = 550(1.03)10
⇒ y = 740 (approx.)
Practice Problems
- Population Growth: A city has a population of 50,000 people, and it grows at a rate of 4% per year. What will the population be like after 10 years?
- Investment Growth: If you invest $1,000 at an annual interest rate of 5%, how much will the investment be worth after 15 years?
- Bacterial Growth: A bacterial culture starts with 200 bacteria and doubles in number every 3 hours. How many bacteria will there be after 12 hours?
- Compound Interest: If you deposit $500 into a savings account with an annual compound interest rate of 3%, how much will be in the account after 8 years?
- Radioactive Decay: A radioactive substance has a half-life of 10 years. If the initial amount is 100 grams, how much will remain after 30 years?
- Debt Repayment: You owe $2,000 on a loan with an annual interest rate of 7%, compounded yearly. How much will you owe after 5 years?
- Population Decline: A town's population is decreasing at a rate of 2% per year. If the current population is 80,000, what will it be in 20 years?
- Viral Spread: A virus infects 50 people initially and spreads such that the number of infected people triples every 4 days. How many people will be infected after 16 days?
- Growth Rate Calculation: If a quantity grows from 1,000 to 2,000 in 5 years, what is the annual growth rate?
- Future Value of Investment: An investment of $10,000 grows to $15,000 in 7 years. What is the annual growth rate?