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Answer :
To solve the problem, we are given the function [tex]\( f(t) = \rho e^t \)[/tex] and specific values for it:
1. [tex]\( f(3) = 191.5 \)[/tex]
2. [tex]\( r = 0.03 \)[/tex]
3. We need to find the approximate value of [tex]\(\rho\)[/tex].
Let's break it down step-by-step:
1. Identify the known values:
- [tex]\( t = 3 \)[/tex]
- [tex]\( f(3) = \rho e^3 = 191.5 \)[/tex]
2. Substitute into the function:
[tex]\[
191.5 = \rho e^3
\][/tex]
3. Solve for [tex]\(\rho\)[/tex]:
- Rearrange the equation to find [tex]\(\rho\)[/tex]:
[tex]\[
\rho = \frac{191.5}{e^3}
\][/tex]
4. Calculate [tex]\(e^3\)[/tex]:
- The value of [tex]\(e^3\)[/tex] is approximately 20.0855.
5. Calculate [tex]\(\rho\)[/tex]:
[tex]\[
\rho = \frac{191.5}{20.0855} \approx 9.53
\][/tex]
6. Find the approximate value of [tex]\(\rho\)[/tex]:
- By rounding [tex]\(\rho\)[/tex], we find that the approximate value is 10.
Therefore, the approximate value of [tex]\(\rho\)[/tex] is closest to 10, which matches none of the given choices. If [tex]\(P\)[/tex] represents this approximate value, it seems there might be an error or discrepancy with the original choices, since none of them exactly match 10.
1. [tex]\( f(3) = 191.5 \)[/tex]
2. [tex]\( r = 0.03 \)[/tex]
3. We need to find the approximate value of [tex]\(\rho\)[/tex].
Let's break it down step-by-step:
1. Identify the known values:
- [tex]\( t = 3 \)[/tex]
- [tex]\( f(3) = \rho e^3 = 191.5 \)[/tex]
2. Substitute into the function:
[tex]\[
191.5 = \rho e^3
\][/tex]
3. Solve for [tex]\(\rho\)[/tex]:
- Rearrange the equation to find [tex]\(\rho\)[/tex]:
[tex]\[
\rho = \frac{191.5}{e^3}
\][/tex]
4. Calculate [tex]\(e^3\)[/tex]:
- The value of [tex]\(e^3\)[/tex] is approximately 20.0855.
5. Calculate [tex]\(\rho\)[/tex]:
[tex]\[
\rho = \frac{191.5}{20.0855} \approx 9.53
\][/tex]
6. Find the approximate value of [tex]\(\rho\)[/tex]:
- By rounding [tex]\(\rho\)[/tex], we find that the approximate value is 10.
Therefore, the approximate value of [tex]\(\rho\)[/tex] is closest to 10, which matches none of the given choices. If [tex]\(P\)[/tex] represents this approximate value, it seems there might be an error or discrepancy with the original choices, since none of them exactly match 10.
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