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The problem provides an exponential model \( f(t) = 2e^{r \cdot t} \), where \( t \) is the number of days since day 0, and \( r \) is the growth rate that we need to determine. The table provides values for \( f(0) = 2 \) grams and \( f(5) = 6.315 \) grams.

### Step-by-Step Solution:

We are given the exponential equation:
\[
f(t) = 2e^{r \cdot t}
\]
- For \( t = 0 \): \( f(0) = 2e^{r \cdot 0} = 2 \times 1 = 2 \) (this is already given, so the equation holds).
- For \( t = 5 \): \( f(5) = 2e^{5r} = 6.315 \).

Now, we can solve for \( r \) by substituting into the equation:
\[
6.315 = 2e^{5r}
\]
First, divide both sides by 2:
\[
e^{5r} = \frac{6.315}{2} = 3.1575
\]

Next, take the natural logarithm (ln) of both sides to eliminate the exponential:
\[
\ln(e^{5r}) = \ln(3.1575)
\]
This simplifies to:
\[
5r = \ln(3.1575)
\]

Using a calculator, we find:
\[
\ln(3.1575) \approx 1.149
\]

Now, solve for \( r \):
\[
r = \frac{1.149}{5} = 0.2298
\]

### Final Answer:
The value of \( r \) is approximately \( 0.230 \), which corresponds to option **A**.

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Would you like more details or further explanation on any part of this solution? Here are some related questions you might find useful:

1. How is the exponential growth model derived from real-life data?
2. What other forms of exponential models exist, and how do they differ from this one?
3. How can we interpret the growth rate \( r \) in the context of this problem?
4. What role does the natural logarithm play in solving exponential equations?
5. How would the solution change if the data values or time periods were different?

**Tip**: When dealing with exponential functions, taking the natural logarithm is a key step to "bring down" the exponent and solve for unknown variables.

The table gives values for the amount of a certain substance, in grams, on certain days. The data are modeled by an exponential function f(t) = 2e^(rt), where t is the number of days since day 0. The constant r is defined as the continuous growth rate of this model. Based on the table, what is the value of r?

Solve an exponential growth problem where the growth rate r is determined from given data. The equation f(t) = 2e^(rt) models the amount of substance in grams over time. Learn how to use logarithms to solve for r based on values from a table. Suitable for students in Grades 10-12.

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