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------------------------------------------------ If [tex]f(4) = 246.4[/tex] when [tex]r = 0.04[/tex] for the function [tex]f(t) = P e^t[/tex], then what is the approximate value of [tex]P[/tex]?

A. 210
B. 289
C. 50
D. 1220

Let's solve the problem step-by-step to find the approximate value of [tex]\( P \)[/tex].

We have the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. We're given that [tex]\( f(4) = 246.4 \)[/tex]. We also know [tex]\( r = 0.04 \)[/tex] and [tex]\( t = 4 \)[/tex].

Our goal is to find the approximate value of [tex]\( P \)[/tex].

### Step 1: Write the Equation for [tex]\( f(4) \)[/tex]

Given:

[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]

### Step 2: Substitute the Known Value

We know [tex]\( f(4) = 246.4 \)[/tex], so:

[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]

### Step 3: Simplify the Exponent

Calculate the exponent:

[tex]\( 0.04 \cdot 4 = 0.16 \)[/tex]

So, the equation becomes:

[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]

### Step 4: Solve for [tex]\( P \)[/tex]

To find [tex]\( P \)[/tex], rearrange the equation:

[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]

### Step 5: Calculate [tex]\( e^{0.16} \)[/tex]

Using a calculator:

[tex]\[
e^{0.16} \approx 1.1735
\][/tex]

### Step 6: Calculate [tex]\( P \)[/tex]

Divide 246.4 by 1.1735:

[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]

### Step 7: Choose the Closest Option

The closest value among the options is [tex]\( P \approx 210 \)[/tex].

So, the approximate value of [tex]\( P \)[/tex] is:

A. 210