To solve the problem of determining the mass of a substance after five days of continuous exponential growth, we follow these steps:
1. Identify the given variables:
- Initial mass ([tex]\(M_0\)[/tex]) = 354 grams
- Growth rate per day ([tex]\(r\)[/tex]) = 9% per day = 0.09 (in decimal form)
- Time period ([tex]\(t\)[/tex]) = 5 days
2. Understand the exponential growth formula:
The formula for continuous exponential growth is given by:
[tex]\[
M(t) = M_0 \cdot e^{rt}
\][/tex]
where:
- [tex]\(M(t)\)[/tex] is the mass after time [tex]\(t\)[/tex],
- [tex]\(M_0\)[/tex] is the initial mass,
- [tex]\(e\)[/tex] is the base of the natural logarithm (approximately equal to 2.71828),
- [tex]\(r\)[/tex] is the growth rate,
- [tex]\(t\)[/tex] is the time period.
3. Substitute the given values into the formula:
[tex]\[
M(5) = 354 \cdot e^{0.09 \cdot 5}
\][/tex]
4. Calculate the exponent:
[tex]\[
0.09 \cdot 5 = 0.45
\][/tex]
5. Evaluate the exponential function:
[tex]\[
e^{0.45} \approx 1.568312185490169
\][/tex]
Note: This is the approximation of the value of [tex]\(e^{0.45}\)[/tex].
6. Calculate the final mass:
[tex]\[
M(5) = 354 \cdot 1.568312185490169 \approx 555.1825136635197 \text{ grams}
\][/tex]
7. Round the final answer to the nearest tenth:
Thus, the mass after five days, rounded to the nearest tenth, is:
[tex]\[
\boxed{555.2 \text{ grams}}
\][/tex]
Therefore, the mass of the sample after five days is approximately 555.2 grams.
