Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{r t} \)[/tex], given that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex], follow these steps:
1. Substitute the Known Values:
We know:
[tex]\[
f(t) = P e^{r t}
\][/tex]
Given:
[tex]\( f(3) = 191.5 \)[/tex]
[tex]\( r = 0.03 \)[/tex]
[tex]\( t = 3 \)[/tex]
Substitute these into the function:
[tex]\[
191.5 = P \times e^{0.03 \times 3}
\][/tex]
2. Calculate the Exponential Term:
First, calculate the exponent:
[tex]\[
0.03 \times 3 = 0.09
\][/tex]
Then, calculate the value of [tex]\( e^{0.09} \)[/tex].
3. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
4. Approximate [tex]\( P \)[/tex]:
After calculating the above expression, you find that the approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex], which corresponds to option B.
1. Substitute the Known Values:
We know:
[tex]\[
f(t) = P e^{r t}
\][/tex]
Given:
[tex]\( f(3) = 191.5 \)[/tex]
[tex]\( r = 0.03 \)[/tex]
[tex]\( t = 3 \)[/tex]
Substitute these into the function:
[tex]\[
191.5 = P \times e^{0.03 \times 3}
\][/tex]
2. Calculate the Exponential Term:
First, calculate the exponent:
[tex]\[
0.03 \times 3 = 0.09
\][/tex]
Then, calculate the value of [tex]\( e^{0.09} \)[/tex].
3. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
4. Approximate [tex]\( P \)[/tex]:
After calculating the above expression, you find that the approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex], which corresponds to option B.
