Answer :
We are given the function
[tex]$$
f(t) = P e^{rt},
$$[/tex]
with [tex]\( r = 0.03 \)[/tex] and [tex]\( f(3) = 191.5 \)[/tex]. Substituting [tex]\( t = 3 \)[/tex] into the function, we have
[tex]$$
191.5 = P e^{0.03 \times 3}.
$$[/tex]
Simplify the exponent:
[tex]$$
0.03 \times 3 = 0.09.
$$[/tex]
So, the equation becomes
[tex]$$
191.5 = P e^{0.09}.
$$[/tex]
To solve for [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^{0.09} \)[/tex]:
[tex]$$
P = \frac{191.5}{e^{0.09}}.
$$[/tex]
Evaluating [tex]\( e^{0.09} \)[/tex] gives approximately [tex]\( 1.09417 \)[/tex], so
[tex]$$
P \approx \frac{191.5}{1.09417} \approx 175.
$$[/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is [tex]\( 175 \)[/tex].
[tex]$$
f(t) = P e^{rt},
$$[/tex]
with [tex]\( r = 0.03 \)[/tex] and [tex]\( f(3) = 191.5 \)[/tex]. Substituting [tex]\( t = 3 \)[/tex] into the function, we have
[tex]$$
191.5 = P e^{0.03 \times 3}.
$$[/tex]
Simplify the exponent:
[tex]$$
0.03 \times 3 = 0.09.
$$[/tex]
So, the equation becomes
[tex]$$
191.5 = P e^{0.09}.
$$[/tex]
To solve for [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^{0.09} \)[/tex]:
[tex]$$
P = \frac{191.5}{e^{0.09}}.
$$[/tex]
Evaluating [tex]\( e^{0.09} \)[/tex] gives approximately [tex]\( 1.09417 \)[/tex], so
[tex]$$
P \approx \frac{191.5}{1.09417} \approx 175.
$$[/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is [tex]\( 175 \)[/tex].
