Answer :
We are given the function
[tex]$$
f(t) = P e^{r t},
$$[/tex]
and the condition
[tex]$$
f(3) = 191.5 \quad \text{with} \quad r = 0.03.
$$[/tex]
Substitute [tex]\( t = 3 \)[/tex] and [tex]\( r = 0.03 \)[/tex] into the function:
[tex]$$
191.5 = P e^{0.03 \times 3} = P e^{0.09}.
$$[/tex]
To solve for [tex]\( P \)[/tex], divide both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]$$
P = \frac{191.5}{e^{0.09}}.
$$[/tex]
Since
[tex]$$
e^{0.09} \approx 1.09417,
$$[/tex]
we have
[tex]$$
P \approx \frac{191.5}{1.09417} \approx 175.
$$[/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex], which corresponds to option C.
[tex]$$
f(t) = P e^{r t},
$$[/tex]
and the condition
[tex]$$
f(3) = 191.5 \quad \text{with} \quad r = 0.03.
$$[/tex]
Substitute [tex]\( t = 3 \)[/tex] and [tex]\( r = 0.03 \)[/tex] into the function:
[tex]$$
191.5 = P e^{0.03 \times 3} = P e^{0.09}.
$$[/tex]
To solve for [tex]\( P \)[/tex], divide both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]$$
P = \frac{191.5}{e^{0.09}}.
$$[/tex]
Since
[tex]$$
e^{0.09} \approx 1.09417,
$$[/tex]
we have
[tex]$$
P \approx \frac{191.5}{1.09417} \approx 175.
$$[/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex], which corresponds to option C.
