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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. 50
B. 210
C. 289
D. 1220

Answer :

To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], we can follow these steps:

1. Substitute Known Values:

We have the equation [tex]\( f(t) = P e^{rt} \)[/tex]. We are given [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex]. So, we substitute these values into the equation:

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

2. Calculate the Exponential Part:

Calculate [tex]\( e^{0.04 \times 4} \)[/tex]. This can be done using a scientific calculator or an exponential function in mathematical software:

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

3. Solve for [tex]\( P \)[/tex]:

Now, solve the equation for [tex]\( P \)[/tex]:

[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]

[tex]\[
P \approx 209.97
\][/tex]

4. Round or Choose the Closest Value:

Looking at the options provided:

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

The calculated value [tex]\( P \approx 209.97 \)[/tex] is closest to option B, 210.

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

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