Exponential Growth

Show your work for each problem.

1.
Which formula represents exponential growth?
1. a)
2. b)
3. c)
4. d)
Answer:
is the exponential growth formula where the initial amount is multiplied by the growth factor raised to the power of time.
2.
If something grows by 12% per year, what is the growth factor ?
Answer: 1.12
The growth factor . The growth rate 12% is converted to decimal form (0.12) and added to 1.
3.
A population doubles every year. What is the growth factor?
1. a)
2. b)
3. c)
4. d)
Answer:
When a quantity doubles, it is multiplied by 2 each period. Therefore, the growth factor .
4.
If , what is ?
Answer: 32
. Each time the exponent increases by 1, we multiply by the base.
5.
A bacteria colony starts with 100 cells and doubles every hour. How many cells after 3 hours?
Answer: 800
1. What is the initial amount ? 100
2. What is the growth factor for doubling? 2
3. What is ? 8
4. Calculate 800
6.
What is the difference between exponential and linear growth?
1. a)Linear multiplies; exponential adds
2. b)Both add the same amount each period
3. c)Linear adds a constant; exponential multiplies by a constant
4. d)Exponential is always faster than linear
Answer: Linear adds a constant; exponential multiplies by a constant
Linear growth: (add constant). Exponential growth: (multiply by constant). The key difference is addition vs multiplication.
7.
Calculate rounded to two decimal places.
Answer: 1.16
. This means 5% growth over 3 years results in about 16% total increase.
8.
You invest 500 euros at 10% annual interest. How much after 2 years?
Answer: 605
1. What is the growth factor for 10% growth? 1.1
2. What is ? 1.21
3. Calculate 605
9.
A quantity triples every year. After 4 years, it will be multiplied by:
1. a)
2. b)
3. c)
4. d)
Answer:
With growth factor and time , the multiplier is . This is NOT - exponential growth uses exponents!
10.
Using the Rule of 70: If an investment grows at 7% per year, approximately how many years to double?
Answer: 10
The Rule of 70 states: Doubling time ≈ years. This is a quick estimation for compound growth.
11.
A town of 20,000 people grows at 5% per year. What's the population after 3 years?
Answer: 23153
1. What is the growth factor? 1.05
2. Calculate 1.157625
3. Calculate (round to nearest whole number) 23153
12.
An investment grew from 1000 to 1210 euros in 2 years. What was the annual growth rate?
1. a)10.5%
2. b)10%
3. c)5%
4. d)21%
Answer: 10%
, so , thus . The growth rate is .
13.
A population grew from 5000 to 8000 in 4 years. Find the annual growth rate (round to 1 decimal).
Answer: 12.5
1. Set up the equation: . What is ? 1.6
2. Find (use calculator) 1.125
3. Convert to percentage: , then multiply by 100 12.5
14.
How many years for 1000 euros to grow to 2000 euros at 10% annual interest? (Round to nearest whole number)
Answer: 7
Using Rule of 70: years. Exact calculation: , so years.
15.
You invest 2000 euros at 8% annually. How much after 5 years? (Round to nearest euro)
Answer: 2939
1. What is the growth factor for 8%? 1.08
2. Calculate 1.469
3. Calculate , round to nearest euro 2939
16.
Why does exponential growth eventually become unsustainable in real-world populations?
1. a)Growth rates always decrease naturally
2. b)The math formula breaks down
3. c)Resources become limited (food, space)
4. d)Numbers get too big to count
Answer: Resources become limited (food, space)
In the real world, resources like food, water, and space are limited. As populations grow, competition increases and growth slows (logistic growth). The exponential model is only accurate for early growth phases.