College

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

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

Answer :

Certainly! Let's solve the question step-by-step:

We are given the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], where we're asked to find the approximate value of [tex]\( P \)[/tex]. We know that:

- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]

We need to determine [tex]\( P \)[/tex] for [tex]\( t = 4 \)[/tex].

### Step 1: Write the equation with known values

Plug the known values into the equation:

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

Since [tex]\( f(4) = 246.4 \)[/tex], the equation becomes:

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

### Step 2: Solve for [tex]\( P \)[/tex]

To find [tex]\( P \)[/tex], we need to isolate it on one side of the equation:

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

### Step 3: Calculate the exponential part

The value of [tex]\( e^{0.16} \)[/tex] is approximately 1.1735.

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

Now, divide 246.4 by 1.1735 to get [tex]\( P \)[/tex]:

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

Therefore, the approximate value of [tex]\( P \)[/tex] is 210.

### Conclusion

The closest answer choice to 209.97 is option D: 210. Hence, the approximate value of [tex]\( P \)[/tex] is 210.