Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = \rho e^{r \cdot t} \)[/tex], given [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex], follow these steps:
1. Understand the function and the values provided:
- The formula for the function is [tex]\( f(t) = \rho e^{r \cdot t} \)[/tex].
- We know [tex]\( f(3) = 191.5 \)[/tex].
- We need to find [tex]\( \rho \)[/tex] (which is labeled as [tex]\( P \)[/tex] in the options).
2. Set up the equation using the given values:
[tex]\[
f(3) = \rho e^{0.03 \cdot 3}
\][/tex]
Substitute the values into the equation:
[tex]\[
191.5 = \rho \times e^{0.09}
\][/tex]
3. Calculate the value of [tex]\( e^{0.09} \)[/tex]:
- Using exponential calculations, [tex]\( e^{0.09} \approx 1.0942 \)[/tex].
4. Solve for [tex]\( \rho \)[/tex]:
- Rearrange the equation:
[tex]\[
\rho = \frac{191.5}{e^{0.09}}
\][/tex]
[tex]\[
\rho = \frac{191.5}{1.0942}
\][/tex]
5. Compute [tex]\( \rho \)[/tex]:
- Perform the division to find [tex]\( \rho \)[/tex]:
[tex]\[
\rho \approx 175.02
\][/tex]
6. Choose the closest option:
- Comparing [tex]\( 175.02 \)[/tex] with the given options, the closest answer is [tex]\( 175 \)[/tex].
Thus, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{175}\)[/tex].
1. Understand the function and the values provided:
- The formula for the function is [tex]\( f(t) = \rho e^{r \cdot t} \)[/tex].
- We know [tex]\( f(3) = 191.5 \)[/tex].
- We need to find [tex]\( \rho \)[/tex] (which is labeled as [tex]\( P \)[/tex] in the options).
2. Set up the equation using the given values:
[tex]\[
f(3) = \rho e^{0.03 \cdot 3}
\][/tex]
Substitute the values into the equation:
[tex]\[
191.5 = \rho \times e^{0.09}
\][/tex]
3. Calculate the value of [tex]\( e^{0.09} \)[/tex]:
- Using exponential calculations, [tex]\( e^{0.09} \approx 1.0942 \)[/tex].
4. Solve for [tex]\( \rho \)[/tex]:
- Rearrange the equation:
[tex]\[
\rho = \frac{191.5}{e^{0.09}}
\][/tex]
[tex]\[
\rho = \frac{191.5}{1.0942}
\][/tex]
5. Compute [tex]\( \rho \)[/tex]:
- Perform the division to find [tex]\( \rho \)[/tex]:
[tex]\[
\rho \approx 175.02
\][/tex]
6. Choose the closest option:
- Comparing [tex]\( 175.02 \)[/tex] with the given options, the closest answer is [tex]\( 175 \)[/tex].
Thus, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{175}\)[/tex].
