Question: 5 — Exponential Decay
LEARNING OBJECTIVE: Calculate the decay time by solving for $x$ in the
exponential equation.
A certain element decays at a constant rate of 3% per year.
If you start with 15 grams of the element, how long will it take before there
are only three grams left?
a.) 52.8 years
b.) 51.8 years
c.) 54.4 years
d.) 4.51 years
1. **State the problem:** We want to find the time $x$ (in years) it takes for 15 grams of an element to decay to 3 grams, given a constant decay rate of 3% per year.
2. **Formula used:** The exponential decay formula is $$A = A_0 \times (1 - r)^x$$ where:
- $A$ is the amount remaining after time $x$,
- $A_0$ is the initial amount,
- $r$ is the decay rate per year,
- $x$ is the time in years.
3. **Plug in known values:**
$$3 = 15 \times (1 - 0.03)^x$$
4. **Simplify the base:**
$$3 = 15 \times 0.97^x$$
5. **Isolate the exponential term:**
$$\frac{3}{15} = 0.97^x$$
6. **Simplify the fraction:**
$$\frac{\cancel{3}}{\cancel{15}} = \frac{1}{5} = 0.2 = 0.97^x$$
7. **Take the natural logarithm of both sides:**
$$\ln(0.2) = \ln(0.97^x)$$
8. **Use logarithm power rule:**
$$\ln(0.2) = x \times \ln(0.97)$$
9. **Solve for $x$:**
$$x = \frac{\ln(0.2)}{\ln(0.97)}$$
10. **Calculate values:**
$$\ln(0.2) \approx -1.6094$$
$$\ln(0.97) \approx -0.03046$$
11. **Divide:**
$$x = \frac{-1.6094}{-0.03046} \approx 52.83$$
12. **Interpretation:** It will take approximately 52.8 years for the element to decay from 15 grams to 3 grams.
**Final answer:** a.) 52.8 years