Exponential Growth and Decay- Model Worksheet Practice
What Exponential Growth and Decay Actually Are
Exponential growth and decay describe how quantities change over time at a rate proportional to their current value. Unlike linear change, where you add or subtract the same amount each time, exponential change multiplies or divides by a factor.
Your money in a savings account? Exponential growth. The caffeine in your system after your morning coffee? Exponential decay. This stuff shows up everywhere, and if you're working through worksheets, you need to actually understand the mechanics—not just memorize steps.
The Formulas You Must Know
Two formulas. That's it. Memorize them.
Exponential Growth
y = a(1 + r)^t
Where:
- a = starting amount
- r = growth rate (as a decimal)
- t = time period
- y = final amount
Exponential Decay
y = a(1 - r)^t
The only difference is the minus sign. Growth adds, decay subtracts. Everything else works the same way.
Growth vs. Decay: Quick Comparison
| Feature | Growth | Decay |
|---|---|---|
| Formula | y = a(1 + r)^t | y = a(1 - r)^t |
| Rate | Positive (r > 0) | Positive (r > 0), subtracted |
| Result | Increases over time | Decreases over time |
| Common examples | Population, investments, compound interest | Radioactive decay, depreciation, drug absorption |
How to Solve These Problems: Step by Step
Most worksheet problems fall into three categories. Here's how to handle each one.
Finding the Final Amount
Plug in what you know. That's literally it.
Example: You invest $1,000 at 5% annual interest for 10 years. What's the final amount?
y = 1000(1 + 0.05)^10
y = 1000(1.6289)
y = $1,628.89
Finding the Growth/Decay Rate
Rearrange the formula. Solve for r.
Example: A bacteria culture grows from 100 to 500 in 3 hours. What's the growth rate?
500 = 100(1 + r)^3
5 = (1 + r)^3
1.71 ≈ 1 + r
r ≈ 0.71 or 71% per hour
Finding the Time
Use logarithms. This is where most students get stuck.
Example: How long to triple $500 at 8% annual interest?
1500 = 500(1.08)^t
3 = (1.08)^t
t × log(1.08) = log(3)
t = log(3) ÷ log(1.08) ≈ 14.2 years
Common Mistakes That Cost You Points
- Confusing the rate: 5% is 0.05, not 5. This one shows up constantly.
- Wrong formula: Using growth when you should use decay. Read the problem carefully.
- Forgetting parentheses: (1.05)^10 is not the same as 1.05^10. The first is correct.
- Logarithm errors: ln vs. log—know when to use each. For base 10, use log. For natural growth, use ln.
- Not checking your answer: Plug it back in. Does it make sense?
Practice Problems to Work Through
Here's what your worksheet is probably asking for. Try these before looking at solutions.
Problem 1: Basic Growth
A city had 50,000 residents in 2020. Population grows at 3% per year. What's the population in 2030?
Answer: y = 50000(1.03)^10 ≈ 67,195
Problem 2: Decay Application
A car worth $25,000 depreciates 12% per year. What's it worth after 5 years?
Answer: y = 25000(0.88)^5 ≈ $13,159
Problem 3: Finding Time
An investment doubles at 6% compound interest. How long does it take?
Answer: t = log(2) ÷ log(1.06) ≈ 11.9 years
Problem 4: Half-Life
A radioactive substance has a half-life of 8 years. If you start with 200 grams, how much remains after 24 years?
Answer: 200(0.5)^3 = 25 grams
Getting Started: Your Worksheet Strategy
Follow this approach for every problem:
- Identify the type. Growth or decay? Look for keywords: "increase," "depreciate," "half-life," "compound."
- Pull out your variables. What is a? What is r? What is t?
- Choose the correct formula. + for growth, - for decay.
- Solve. Use algebra or logarithms as needed.
- Check your work. Does the number make sense? Is the growth/decay direction correct?
This process works every time. No guessing, no panicking.
When You're Stuck: What to Do
If a problem has you frozen:
- Draw a before-and-after picture. Sketch the starting and ending values.
- Write down what you know. Fill in the formula with numbers you have.
- Ask yourself what's missing. That's your variable to solve for.
- Work backwards. If the answer should be smaller and you got bigger, flip your sign.
Most mistakes come from rushing. Slow down on the setup, and the math takes care of itself.
What Comes Next
Once you can handle these problems consistently, you'll see them in calculus (derivatives of exponential functions), finance (present value calculations), and science (modeling real systems). The worksheets are building blocks, not busywork.
Work through your problems. Check your answers. Fix what you got wrong. That's the entire process.