Exponential Decay and Growth- Comprehensive Worksheet
What Is Exponential Decay and Growth?
Exponential decay and growth describe processes where quantities change by a constant multiplicative rate over equal time intervals. Unlike linear change, where you add or subtract the same amount each step, exponential change means you multiply or divide by the same factor.
This isn't abstract math. It shows up in:
- Radioactive materials breaking down over time
- Population growth in biology
- Compound interest in finance
- Cooling coffee on your desk
- Drug concentration in your bloodstream
If you're working with a worksheet on this topic, you need three things: the right formulas, the ability to identify which type of problem you're dealing with, and practice applying the steps correctly.
The Core Formulas You Must Know
Exponential Growth
Formula: y = a(1 + r)t
Where:
- a = initial amount
- r = growth rate (as a decimal)
- t = time period
- y = final amount
The factor (1 + r) is your growth multiplier. A 5% growth rate means you multiply by 1.05 each period.
Exponential Decay
Formula: y = a(1 - r)t
Same structure, but now (1 - r) is your decay multiplier. A 12% decay rate means you multiply by 0.88 each period.
The General Form
Sometimes you'll see this:
y = abt
Where b is your multiplier. Growth = b > 1. Decay = 0 < b < 1.
That's it. Memorize these or write them down before starting the worksheet.
Growth vs. Decay: How to Tell Them Apart
Your worksheet will usually tell you which type applies. But if it doesn't:
- Is something increasing over time? Growth.
- Is something decreasing over time? Decay.
- Is the rate positive? Growth. Negative rate? Decay.
Don't overthink this. The problem context usually makes it obvious.
Comparing Growth and Decay
| Feature | Exponential Growth | Exponential Decay |
|---|---|---|
| Multiplier (b) | b > 1 | 0 < b < 1 |
| Rate (r) | Positive (added to 1) | Subtracted from 1 |
| Graph shape | Rises, gets steeper | Falls, flattens out |
| Asymptote | No upper limit | Approaches zero |
| Common examples | Populations, investments | Radioactive decay, cooling |
How to Solve Exponential Problems: Step by Step
Here's the process for any worksheet problem:
Step 1: Identify What You're Solving For
Read the problem. Circle the initial amount (a), the rate (r), the time (t), and what you're solving for (y or t).
Step 2: Pick the Right Formula
Growth = y = a(1 + r)t
Decay = y = a(1 - r)t
Step 3: Plug In the Numbers
Substitute your known values. Make sure the rate is a decimal, not a percentage. 8% = 0.08, not 8.
Step 4: Calculate
Use your calculator for the exponent. Work inside parentheses first.
Step 5: Check Your Work
Does your answer make sense? Growth problems should give you a larger number than you started with. Decay problems should give you a smaller number.
Practice Problems for Your Worksheet
Work through these. Cover the answers, try them yourself, then check.
Problem 1: A bacteria population starts at 500 and grows at 3% per hour. What is the population after 4 hours?
Solution:
- a = 500
- r = 0.03
- t = 4
- y = 500(1.03)4
- y = 500 × 1.1255
- y ≈ 563 bacteria
Problem 2: A radioactive sample has 200 grams and decays at 6% per year. How much remains after 10 years?
Solution:
- a = 200
- r = 0.06
- t = 10
- y = 200(0.94)10
- y = 200 × 0.5386
- y ≈ 107.7 grams
Problem 3: You invest $1,000 at 7% compound interest. How long until you reach $1,500?
Solution:
- a = 1000
- y = 1500
- r = 0.07
- 1500 = 1000(1.07)t
- 1.5 = (1.07)t
- t = log(1.5) / log(1.07)
- t ≈ 5.99 years ≈ 6 years
Common Mistakes That Cost You Points
- Using addition instead of multiplication. Exponential means multiply by the factor each period, not add to it.
- Forgetting to convert percentages to decimals. 15% = 0.15. Always.
- Getting growth and decay backwards. If something is dying off, leaking out, or cooling down, it's decay. Not growth.
- Rounding too early. Keep full decimal values until your final answer.
- Ignoring the exponent. (1.05)10 is not the same as 1.05 × 10.
Real-World Applications
Your teacher assigned this worksheet for a reason. These formulas actually get used:
- Medicine: Half-life calculations for drug dosing intervals
- Archaeology: Carbon-14 dating uses exponential decay to determine fossil ages
- Finance: Every compound interest calculation uses exponential growth
- Physics: Nuclear decay rates and half-life of isotopes
- Environmental science: Modeling pollutant breakdown in ecosystems
Solving for Time (t) in Exponential Equations
Sometimes your worksheet asks you to find when something happens, not what the amount is. This requires logarithms.
The general approach:
- Isolate the exponential term on one side
- Take the log of both sides
- Use the log power rule: log(bt) = t × log(b)
- Solve for t
Example: When does a $200 investment at 5% compound interest become $500?
500 = 200(1.05)t
2.5 = (1.05)t
log(2.5) = t × log(1.05)
t = log(2.5) / log(1.05) ≈ 18.78 years
Half-Life Problems
Half-life is a specific type of decay problem. The half-life is the time it takes for a quantity to halve.
Formula: y = a(0.5)t/h
Where h = half-life period.
Example: A substance has a half-life of 3 years. Starting with 80 grams, what's left after 9 years?
- a = 80
- h = 3
- t = 9
- y = 80(0.5)9/3
- y = 80(0.5)3
- y = 80 × 0.125
- y = 10 grams
Getting Started on Your Worksheet
Before you begin:
- Write down both formulas. Keep them visible.
- Read each problem twice before solving.
- Identify whether it's growth or decay first.
- Convert all percentages to decimals.
- Show your work. Teachers grade the process, not just the answer.
- Use a calculator for exponents. Know how to use the log and ln buttons.
If you're stuck on a problem, work backwards. If you can set up the equation correctly, the math takes care of itself.
Quick Reference
| What You're Solving | Formula |
|---|---|
| Final amount (growth) | y = a(1 + r)t |
| Final amount (decay) | y = a(1 - r)t |
| Time (known final amount) | t = log(y/a) / log(1 ± r) |
| Half-life problems | y = a(0.5)t/h |
| General exponential | y = abt |
That's the whole worksheet covered. The formulas are straightforward. The only thing that trips most students up is rushing through the setup. Slow down, identify your variables, plug in correctly, and calculate carefully.