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

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

Answer :

To solve for [tex]\( P \)[/tex], we know from the problem that:

[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]

We're given:

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

We need to find [tex]\( P \)[/tex].

1. Substitute the values into the function:

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

2. Simplify the exponent:

[tex]\[ e^{0.04 \cdot 4} = e^{0.16} \][/tex]

3. Calculate [tex]\( e^{0.16} \)[/tex].

4. Solve for [tex]\( P \)[/tex] by dividing both sides of the equation by [tex]\( e^{0.16} \)[/tex]:

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

Using the above steps, the approximate value of [tex]\( P \)[/tex] is found to be about 210.

Therefore, the correct answer is B. 210.

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