Answer :
We are given that
[tex]$$
f(t) = P e^{rt}
$$[/tex]
and that
[tex]$$
f(5) = 288.9
$$[/tex]
with
[tex]$$
r = 0.05.
$$[/tex]
Step 1. Substitute the given values into the function:
[tex]$$
f(5) = P e^{0.05 \times 5}.
$$[/tex]
Step 2. Calculate the exponent:
[tex]$$
0.05 \times 5 = 0.25.
$$[/tex]
So the equation becomes:
[tex]$$
288.9 = P e^{0.25}.
$$[/tex]
Step 3. Solve for [tex]$P$[/tex] by dividing both sides by [tex]$e^{0.25}$[/tex]:
[tex]$$
P = \frac{288.9}{e^{0.25}}.
$$[/tex]
Step 4. Calculate the value of [tex]$e^{0.25}$[/tex]:
[tex]$$
e^{0.25} \approx 1.2840.
$$[/tex]
Step 5. Substitute this value into the equation for [tex]$P$[/tex]:
[tex]$$
P \approx \frac{288.9}{1.2840} \approx 225.
$$[/tex]
Thus, the approximate value of [tex]$P$[/tex] is [tex]$\boxed{225}$[/tex].
[tex]$$
f(t) = P e^{rt}
$$[/tex]
and that
[tex]$$
f(5) = 288.9
$$[/tex]
with
[tex]$$
r = 0.05.
$$[/tex]
Step 1. Substitute the given values into the function:
[tex]$$
f(5) = P e^{0.05 \times 5}.
$$[/tex]
Step 2. Calculate the exponent:
[tex]$$
0.05 \times 5 = 0.25.
$$[/tex]
So the equation becomes:
[tex]$$
288.9 = P e^{0.25}.
$$[/tex]
Step 3. Solve for [tex]$P$[/tex] by dividing both sides by [tex]$e^{0.25}$[/tex]:
[tex]$$
P = \frac{288.9}{e^{0.25}}.
$$[/tex]
Step 4. Calculate the value of [tex]$e^{0.25}$[/tex]:
[tex]$$
e^{0.25} \approx 1.2840.
$$[/tex]
Step 5. Substitute this value into the equation for [tex]$P$[/tex]:
[tex]$$
P \approx \frac{288.9}{1.2840} \approx 225.
$$[/tex]
Thus, the approximate value of [tex]$P$[/tex] is [tex]$\boxed{225}$[/tex].
