Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] with the given condition [tex]\( f(3) = 191.5 \)[/tex] and the rate [tex]\( r = 0.03 \)[/tex], we follow these steps:
1. Understand the Problem: You are given a function [tex]\( f(t) = P e^{rt} \)[/tex], where [tex]\( P \)[/tex] is the initial value or principal, [tex]\( e \)[/tex] is the base of the natural logarithm, [tex]\( r \)[/tex] is the rate, and [tex]\( t \)[/tex] is time. We have to find [tex]\( P \)[/tex] when [tex]\( f(3) = 191.5 \)[/tex].
2. Set Up the Equation: According to the problem:
[tex]\[
f(3) = P e^{r \cdot 3} = 191.5
\][/tex]
3. Plug in the Known Values:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
4. Simplify the Exponent: Calculate the exponent:
[tex]\[
0.03 \times 3 = 0.09
\][/tex]
5. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
6. Approximate the Exponential:
- Calculate [tex]\( e^{0.09} \)[/tex] (this normally involves using a calculator or known exponential values).
7. Calculate [tex]\( P \)[/tex]: After calculating [tex]\( e^{0.09} \)[/tex], divide [tex]\( 191.5 \)[/tex] by this value to find [tex]\( P \)[/tex].
8. Compare with Options: Once you have the approximate value of [tex]\( P \)[/tex], compare it with the given options to find the closest match.
The calculation shows that the approximate value of [tex]\( P \)[/tex] is 175. Therefore, the correct answer among the options is:
A. 175
1. Understand the Problem: You are given a function [tex]\( f(t) = P e^{rt} \)[/tex], where [tex]\( P \)[/tex] is the initial value or principal, [tex]\( e \)[/tex] is the base of the natural logarithm, [tex]\( r \)[/tex] is the rate, and [tex]\( t \)[/tex] is time. We have to find [tex]\( P \)[/tex] when [tex]\( f(3) = 191.5 \)[/tex].
2. Set Up the Equation: According to the problem:
[tex]\[
f(3) = P e^{r \cdot 3} = 191.5
\][/tex]
3. Plug in the Known Values:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
4. Simplify the Exponent: Calculate the exponent:
[tex]\[
0.03 \times 3 = 0.09
\][/tex]
5. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
6. Approximate the Exponential:
- Calculate [tex]\( e^{0.09} \)[/tex] (this normally involves using a calculator or known exponential values).
7. Calculate [tex]\( P \)[/tex]: After calculating [tex]\( e^{0.09} \)[/tex], divide [tex]\( 191.5 \)[/tex] by this value to find [tex]\( P \)[/tex].
8. Compare with Options: Once you have the approximate value of [tex]\( P \)[/tex], compare it with the given options to find the closest match.
The calculation shows that the approximate value of [tex]\( P \)[/tex] is 175. Therefore, the correct answer among the options is:
A. 175
