Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^t \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex], let's follow these steps:
1. Understand the Function:
The function is [tex]\( f(t) = P e^{rt} \)[/tex]. In this case, [tex]\( t = 3 \)[/tex], so we have:
[tex]\[
f(3) = P e^{3r}
\][/tex]
2. Substitute the Given Values:
We know [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex]. Substituting these into the equation, we get:
[tex]\[
191.5 = P e^{3 \times 0.03}
\][/tex]
3. Calculate [tex]\( e^{3 \times 0.03} \)[/tex]:
First, calculate the exponent:
[tex]\[
3 \times 0.03 = 0.09
\][/tex]
Then find [tex]\( e^{0.09} \)[/tex]. This value is approximately 1.0942.
4. Solve for [tex]\( P \)[/tex]:
Next, rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
[tex]\[
P \approx \frac{191.5}{1.0942}
\][/tex]
[tex]\[
P \approx 175.02
\][/tex]
5. Choose the Closest Approximate Value:
The value of [tex]\( P \)[/tex] we calculated is approximately 175.02, which is closest to the option:
[tex]\[
\boxed{175}
\][/tex]
So, the correct choice is A. 175.
1. Understand the Function:
The function is [tex]\( f(t) = P e^{rt} \)[/tex]. In this case, [tex]\( t = 3 \)[/tex], so we have:
[tex]\[
f(3) = P e^{3r}
\][/tex]
2. Substitute the Given Values:
We know [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex]. Substituting these into the equation, we get:
[tex]\[
191.5 = P e^{3 \times 0.03}
\][/tex]
3. Calculate [tex]\( e^{3 \times 0.03} \)[/tex]:
First, calculate the exponent:
[tex]\[
3 \times 0.03 = 0.09
\][/tex]
Then find [tex]\( e^{0.09} \)[/tex]. This value is approximately 1.0942.
4. Solve for [tex]\( P \)[/tex]:
Next, rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
[tex]\[
P \approx \frac{191.5}{1.0942}
\][/tex]
[tex]\[
P \approx 175.02
\][/tex]
5. Choose the Closest Approximate Value:
The value of [tex]\( P \)[/tex] we calculated is approximately 175.02, which is closest to the option:
[tex]\[
\boxed{175}
\][/tex]
So, the correct choice is A. 175.
