Question: 4 — Exponential Growth
LEARNING OBJECTIVE: Calculate a growth amount by solving for $y$ in the
exponential equation.
A colony of bacteria is growing at a rate of 50% per hour.
If this rate of growth remains the same and the colony starts with 100
bacteria, approximately how many bacteria will there be after 6 hours?
a.) 134 bacteria
b.) 2081 bacteria
c.) 1139 bacteria
d.) 1048 bacteria
1. **State the problem:** We want to find the number of bacteria after 6 hours given an initial amount and a growth rate.
2. **Formula:** The exponential growth formula is $$y = y_0 \times (1 + r)^t$$ where:
- $y$ is the amount after time $t$
- $y_0$ is the initial amount
- $r$ is the growth rate per time period (as a decimal)
- $t$ is the number of time periods
3. **Identify values:**
- $y_0 = 100$ bacteria
- $r = 0.50$ (50% growth rate)
- $t = 6$ hours
4. **Calculate:**
$$y = 100 \times (1 + 0.50)^6 = 100 \times 1.5^6$$
5. **Evaluate $1.5^6$ step-by-step:**
$$1.5^2 = 2.25$$
$$1.5^3 = 1.5 \times 2.25 = 3.375$$
$$1.5^4 = 1.5 \times 3.375 = 5.0625$$
$$1.5^5 = 1.5 \times 5.0625 = 7.59375$$
$$1.5^6 = 1.5 \times 7.59375 = 11.390625$$
6. **Multiply by initial amount:**
$$y = 100 \times 11.390625 = 1139.0625$$
7. **Interpretation:** After 6 hours, there will be approximately 1139 bacteria.
**Answer:** c.) 1139 bacteria