Answer :
To solve the problem, we need to find the approximate value of [tex]\( P \)[/tex] for the given function [tex]\( f(t) = P e^t \)[/tex], where we know [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
Step 1: Understand the function provided.
The function given is [tex]\( f(t) = P e^t \)[/tex]. Here [tex]\( P \)[/tex] is a constant we are trying to find, [tex]\( e \)[/tex] is Euler's number (approximately 2.71828), and [tex]\( t \)[/tex] is the time, which in this case is 3 years.
Step 2: Set up the equation using given values.
We are provided that [tex]\( f(3) = 191.5 \)[/tex]. So:
[tex]\[ 191.5 = P e^3 \][/tex]
Step 3: Solve for [tex]\( P \)[/tex].
To find [tex]\( P \)[/tex], rearrange the equation:
[tex]\[ P = \frac{191.5}{e^3} \][/tex]
Step 4: Calculate [tex]\( e^3 \)[/tex].
The approximate value of [tex]\( e^3 \)[/tex] is 20.0855.
Step 5: Divide 191.5 by this value to find [tex]\( P \)[/tex].
[tex]\[ P = \frac{191.5}{20.0855} \approx 9.534 \][/tex]
Given the choices, none of them match exactly. However, we should re-evaluate our options or check our understanding of how [tex]\( r \)[/tex] would have been used if needed (though in this problem, it wasn't utilized).
It seems there might be a misunderstanding or mismatch with choices available, so my recommendation is to re-check the problem settings or context where options are given.
Step 1: Understand the function provided.
The function given is [tex]\( f(t) = P e^t \)[/tex]. Here [tex]\( P \)[/tex] is a constant we are trying to find, [tex]\( e \)[/tex] is Euler's number (approximately 2.71828), and [tex]\( t \)[/tex] is the time, which in this case is 3 years.
Step 2: Set up the equation using given values.
We are provided that [tex]\( f(3) = 191.5 \)[/tex]. So:
[tex]\[ 191.5 = P e^3 \][/tex]
Step 3: Solve for [tex]\( P \)[/tex].
To find [tex]\( P \)[/tex], rearrange the equation:
[tex]\[ P = \frac{191.5}{e^3} \][/tex]
Step 4: Calculate [tex]\( e^3 \)[/tex].
The approximate value of [tex]\( e^3 \)[/tex] is 20.0855.
Step 5: Divide 191.5 by this value to find [tex]\( P \)[/tex].
[tex]\[ P = \frac{191.5}{20.0855} \approx 9.534 \][/tex]
Given the choices, none of them match exactly. However, we should re-evaluate our options or check our understanding of how [tex]\( r \)[/tex] would have been used if needed (though in this problem, it wasn't utilized).
It seems there might be a misunderstanding or mismatch with choices available, so my recommendation is to re-check the problem settings or context where options are given.
