Answer :
We are given a function
[tex]$$
f(t) = P e^{rt},
$$[/tex]
with [tex]$r = 0.05$[/tex]. For [tex]$t = 5$[/tex], the function value is
[tex]$$
f(5) = P e^{0.05 \cdot 5} = 288.9.
$$[/tex]
### Step 1: Simplify the Exponent
First, compute the exponent for [tex]$t = 5$[/tex]:
[tex]$$
0.05 \cdot 5 = 0.25.
$$[/tex]
So the equation becomes:
[tex]$$
P e^{0.25} = 288.9.
$$[/tex]
### Step 2: Solve for [tex]$P$[/tex]
Isolate [tex]$P$[/tex] by dividing both sides by [tex]$e^{0.25}$[/tex]:
[tex]$$
P = \frac{288.9}{e^{0.25}}.
$$[/tex]
### Step 3: Compute the Value of [tex]$e^{0.25}$[/tex]
The value of [tex]$e^{0.25}$[/tex] is approximately
[tex]$$
e^{0.25} \approx 1.2840254166877414.
$$[/tex]
### Step 4: Calculate [tex]$P$[/tex]
Substitute the value of [tex]$e^{0.25}$[/tex] into the equation:
[tex]$$
P = \frac{288.9}{1.2840254166877414} \approx 224.99554622932885.
$$[/tex]
Rounding this value to the nearest whole number, we get approximately [tex]$225$[/tex].
### Conclusion
The approximate value of [tex]$P$[/tex] is [tex]$225$[/tex], which corresponds to option C.
[tex]$$
f(t) = P e^{rt},
$$[/tex]
with [tex]$r = 0.05$[/tex]. For [tex]$t = 5$[/tex], the function value is
[tex]$$
f(5) = P e^{0.05 \cdot 5} = 288.9.
$$[/tex]
### Step 1: Simplify the Exponent
First, compute the exponent for [tex]$t = 5$[/tex]:
[tex]$$
0.05 \cdot 5 = 0.25.
$$[/tex]
So the equation becomes:
[tex]$$
P e^{0.25} = 288.9.
$$[/tex]
### Step 2: Solve for [tex]$P$[/tex]
Isolate [tex]$P$[/tex] by dividing both sides by [tex]$e^{0.25}$[/tex]:
[tex]$$
P = \frac{288.9}{e^{0.25}}.
$$[/tex]
### Step 3: Compute the Value of [tex]$e^{0.25}$[/tex]
The value of [tex]$e^{0.25}$[/tex] is approximately
[tex]$$
e^{0.25} \approx 1.2840254166877414.
$$[/tex]
### Step 4: Calculate [tex]$P$[/tex]
Substitute the value of [tex]$e^{0.25}$[/tex] into the equation:
[tex]$$
P = \frac{288.9}{1.2840254166877414} \approx 224.99554622932885.
$$[/tex]
Rounding this value to the nearest whole number, we get approximately [tex]$225$[/tex].
### Conclusion
The approximate value of [tex]$P$[/tex] is [tex]$225$[/tex], which corresponds to option C.
