Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{e^t} \)[/tex] given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex]. However, the value of [tex]\( r \)[/tex] is not needed for our calculations.
Here's a step-by-step breakdown:
1. Understand the Function: The function is given as [tex]\( f(t) = P e^{e^t} \)[/tex]. We are tasked with finding [tex]\( P \)[/tex].
2. Substitute Known Values: We know that [tex]\( f(4) = 246.4 \)[/tex]. Therefore:
[tex]\[
246.4 = P \, e^{e^4}
\][/tex]
3. Calculate [tex]\( e^4 \)[/tex]: The value [tex]\( e^4 \)[/tex] is the natural exponential function with exponent 4.
4. Calculate [tex]\( e^{e^4} \)[/tex]: You need to exponentiate the result from the previous step, which involves another exponential function.
5. Solve for [tex]\( P \)[/tex]: Since [tex]\( 246.4 = P \cdot e^{e^4} \)[/tex], solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( e^{e^4} \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{e^4}}
\][/tex]
6. Calculate the Result: Upon evaluating [tex]\( e^{e^4} \)[/tex], you find that [tex]\( e^{e^4} \)[/tex] is a very large number. When you divide [tex]\( 246.4 \)[/tex] by this large number, [tex]\( P \)[/tex] turns out to be a very small number in scientific notation:
[tex]\[
P \approx 4.785919858612588 \times 10^{-22}
\][/tex]
Based on this calculation, [tex]\( P \)[/tex] does not match any of the options given: 289, 50, 210, or 1220. This indicates that the options may have been intended for a different set of conditions or calculations. Nonetheless, [tex]\( P \)[/tex] is approximately [tex]\( 4.786 \times 10^{-22} \)[/tex].
Here's a step-by-step breakdown:
1. Understand the Function: The function is given as [tex]\( f(t) = P e^{e^t} \)[/tex]. We are tasked with finding [tex]\( P \)[/tex].
2. Substitute Known Values: We know that [tex]\( f(4) = 246.4 \)[/tex]. Therefore:
[tex]\[
246.4 = P \, e^{e^4}
\][/tex]
3. Calculate [tex]\( e^4 \)[/tex]: The value [tex]\( e^4 \)[/tex] is the natural exponential function with exponent 4.
4. Calculate [tex]\( e^{e^4} \)[/tex]: You need to exponentiate the result from the previous step, which involves another exponential function.
5. Solve for [tex]\( P \)[/tex]: Since [tex]\( 246.4 = P \cdot e^{e^4} \)[/tex], solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( e^{e^4} \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{e^4}}
\][/tex]
6. Calculate the Result: Upon evaluating [tex]\( e^{e^4} \)[/tex], you find that [tex]\( e^{e^4} \)[/tex] is a very large number. When you divide [tex]\( 246.4 \)[/tex] by this large number, [tex]\( P \)[/tex] turns out to be a very small number in scientific notation:
[tex]\[
P \approx 4.785919858612588 \times 10^{-22}
\][/tex]
Based on this calculation, [tex]\( P \)[/tex] does not match any of the options given: 289, 50, 210, or 1220. This indicates that the options may have been intended for a different set of conditions or calculations. Nonetheless, [tex]\( P \)[/tex] is approximately [tex]\( 4.786 \times 10^{-22} \)[/tex].
