Answer :
We are given the function
[tex]$$
f(t) = P\, e^{rt}
$$[/tex]
with [tex]$r = 0.03$[/tex]. Also, it is known that
[tex]$$
f(3) = 191.5.
$$[/tex]
This means that
[tex]$$
191.5 = P\, e^{0.03 \times 3}.
$$[/tex]
Step 1. Calculate the exponent:
[tex]$$
0.03 \times 3 = 0.09.
$$[/tex]
Step 2. Substitute the value back into the equation:
[tex]$$
191.5 = P\, e^{0.09}.
$$[/tex]
Step 3. Solve for [tex]$P$[/tex] by isolating it:
[tex]$$
P = \frac{191.5}{e^{0.09}}.
$$[/tex]
Step 4. Using the approximation [tex]$e^{0.09}\approx 1.09417$[/tex], we compute
[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]$$
f(t) = P\, e^{rt}
$$[/tex]
with [tex]$r = 0.03$[/tex]. Also, it is known that
[tex]$$
f(3) = 191.5.
$$[/tex]
This means that
[tex]$$
191.5 = P\, e^{0.03 \times 3}.
$$[/tex]
Step 1. Calculate the exponent:
[tex]$$
0.03 \times 3 = 0.09.
$$[/tex]
Step 2. Substitute the value back into the equation:
[tex]$$
191.5 = P\, e^{0.09}.
$$[/tex]
Step 3. Solve for [tex]$P$[/tex] by isolating it:
[tex]$$
P = \frac{191.5}{e^{0.09}}.
$$[/tex]
Step 4. Using the approximation [tex]$e^{0.09}\approx 1.09417$[/tex], we compute
[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.
