Answer :
To solve for the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex], you can follow these steps:
1. Substitute the given values into the function equation:
We have the equation [tex]\( f(t) = P e^{rt} \)[/tex]. Substituting the given values, we get:
[tex]\[
191.5 = P \times e^{0.03 \times 3}
\][/tex]
2. Calculate the exponent:
First, calculate the value of [tex]\( r \times t \)[/tex] where [tex]\( r = 0.03 \)[/tex] and [tex]\( t = 3 \)[/tex]:
[tex]\[
r \times t = 0.03 \times 3 = 0.09
\][/tex]
3. Evaluate the exponential part:
Calculate [tex]\( e^{0.09} \)[/tex]. Using the mathematical constant [tex]\( e \approx 2.718 \)[/tex], find the value:
[tex]\[
e^{0.09} \approx 1.0942
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.0942}
\][/tex]
5. Calculate [tex]\( P \)[/tex]:
Perform the division:
[tex]\[
P \approx \frac{191.5}{1.0942} \approx 175.02
\][/tex]
The approximate value of [tex]\( P \)[/tex] is close to 175. Therefore, the correct answer is [tex]\( \boxed{175} \)[/tex].
1. Substitute the given values into the function equation:
We have the equation [tex]\( f(t) = P e^{rt} \)[/tex]. Substituting the given values, we get:
[tex]\[
191.5 = P \times e^{0.03 \times 3}
\][/tex]
2. Calculate the exponent:
First, calculate the value of [tex]\( r \times t \)[/tex] where [tex]\( r = 0.03 \)[/tex] and [tex]\( t = 3 \)[/tex]:
[tex]\[
r \times t = 0.03 \times 3 = 0.09
\][/tex]
3. Evaluate the exponential part:
Calculate [tex]\( e^{0.09} \)[/tex]. Using the mathematical constant [tex]\( e \approx 2.718 \)[/tex], find the value:
[tex]\[
e^{0.09} \approx 1.0942
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.0942}
\][/tex]
5. Calculate [tex]\( P \)[/tex]:
Perform the division:
[tex]\[
P \approx \frac{191.5}{1.0942} \approx 175.02
\][/tex]
The approximate value of [tex]\( P \)[/tex] is close to 175. Therefore, the correct answer is [tex]\( \boxed{175} \)[/tex].
