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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. 289
B. 210
C. 1220
D. 50

Answer :

We are given the function

[tex]$$
f(t) = P e^{rt}
$$[/tex]

and the information that

[tex]$$
f(4) = 246.4 \quad \text{with} \quad r = 0.04.
$$[/tex]

This means that

[tex]$$
P e^{0.04 \times 4} = 246.4.
$$[/tex]

1. First, compute the exponent:

[tex]$$
0.04 \times 4 = 0.16.
$$[/tex]

2. Substitute the exponent back into the equation:

[tex]$$
P e^{0.16} = 246.4.
$$[/tex]

3. Solve for [tex]$P$[/tex] by dividing both sides by [tex]$e^{0.16}$[/tex]:

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

4. Evaluating [tex]$e^{0.16}$[/tex] gives approximately:

[tex]$$
e^{0.16} \approx 1.17351.
$$[/tex]

5. Now, substitute this value into the equation:

[tex]$$
P \approx \frac{246.4}{1.17351} \approx 209.97.
$$[/tex]

This value is nearly [tex]$210$[/tex], so the approximate value of [tex]$P$[/tex] is [tex]$210$[/tex]. Therefore, the correct answer is option B.

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