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Answer :
We are given the function
[tex]$$
f(t) = P \, e^{rt},
$$[/tex]
with [tex]$r = 0.05$[/tex] and [tex]$f(5) = 288.9$[/tex]. First, substitute [tex]$t = 5$[/tex] and [tex]$r = 0.05$[/tex] into the function:
[tex]$$
f(5) = P \, e^{0.05 \times 5} = P \, e^{0.25}.
$$[/tex]
We obtain the equation
[tex]$$
288.9 = P \, e^{0.25}.
$$[/tex]
To solve for [tex]$P$[/tex], we divide both sides by [tex]$e^{0.25}$[/tex]:
[tex]$$
P = \frac{288.9}{e^{0.25}}.
$$[/tex]
It is known that
[tex]$$
e^{0.25} \approx 1.2840254166877414.
$$[/tex]
Thus,
[tex]$$
P \approx \frac{288.9}{1.2840254166877414} \approx 224.99554622932885.
$$[/tex]
Rounded to the nearest whole number, [tex]$P$[/tex] is approximately [tex]$225$[/tex].
Therefore, the correct answer is [tex]$\boxed{225}$[/tex].
[tex]$$
f(t) = P \, e^{rt},
$$[/tex]
with [tex]$r = 0.05$[/tex] and [tex]$f(5) = 288.9$[/tex]. First, substitute [tex]$t = 5$[/tex] and [tex]$r = 0.05$[/tex] into the function:
[tex]$$
f(5) = P \, e^{0.05 \times 5} = P \, e^{0.25}.
$$[/tex]
We obtain the equation
[tex]$$
288.9 = P \, e^{0.25}.
$$[/tex]
To solve for [tex]$P$[/tex], we divide both sides by [tex]$e^{0.25}$[/tex]:
[tex]$$
P = \frac{288.9}{e^{0.25}}.
$$[/tex]
It is known that
[tex]$$
e^{0.25} \approx 1.2840254166877414.
$$[/tex]
Thus,
[tex]$$
P \approx \frac{288.9}{1.2840254166877414} \approx 224.99554622932885.
$$[/tex]
Rounded to the nearest whole number, [tex]$P$[/tex] is approximately [tex]$225$[/tex].
Therefore, the correct answer is [tex]$\boxed{225}$[/tex].
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