Answer :
To solve this problem, we need to find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{r \cdot t} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
Here's how to solve it step by step:
1. Understand the Function:
The function given is [tex]\( f(t) = P e^{r \cdot t} \)[/tex]. We need to find [tex]\( P \)[/tex] when [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex].
2. Set up the Equation:
Plug the given values into the function:
[tex]\[
f(3) = P \cdot e^{0.03 \cdot 3}
\][/tex]
We know [tex]\( f(3) = 191.5 \)[/tex], so:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
3. Calculate [tex]\( e^{0.09} \)[/tex]:
The value of [tex]\( e^{0.09} \)[/tex] is approximately 1.0942. This is a critical calculation step as it will allow solving for [tex]\( P \)[/tex].
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.0942}
\][/tex]
Thus, [tex]\( P \approx 175.02 \)[/tex].
5. Choose the Closest Answer:
From the given options, the closest value to 175.02 is 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{175}\)[/tex].
Here's how to solve it step by step:
1. Understand the Function:
The function given is [tex]\( f(t) = P e^{r \cdot t} \)[/tex]. We need to find [tex]\( P \)[/tex] when [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex].
2. Set up the Equation:
Plug the given values into the function:
[tex]\[
f(3) = P \cdot e^{0.03 \cdot 3}
\][/tex]
We know [tex]\( f(3) = 191.5 \)[/tex], so:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
3. Calculate [tex]\( e^{0.09} \)[/tex]:
The value of [tex]\( e^{0.09} \)[/tex] is approximately 1.0942. This is a critical calculation step as it will allow solving for [tex]\( P \)[/tex].
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.0942}
\][/tex]
Thus, [tex]\( P \approx 175.02 \)[/tex].
5. Choose the Closest Answer:
From the given options, the closest value to 175.02 is 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{175}\)[/tex].
