Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{t^t} \)[/tex] given that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex], follow these steps:
1. Understand the Function and Given Values:
The function given is [tex]\( f(t) = P e^{t^t} \)[/tex]. We need to substitute the values [tex]\( t = 4 \)[/tex], and [tex]\( f(4) = 246.4 \)[/tex].
2. Substitute the Known Values in the Function:
By substituting [tex]\( t = 4 \)[/tex] into the function, we have:
[tex]\[
246.4 = P \cdot e^{4^4}
\][/tex]
3. Calculate [tex]\( 4^4 \)[/tex]:
First, compute the power:
[tex]\[
4^4 = 256
\][/tex]
4. Calculate [tex]\( e^{256} \)[/tex]:
Next, calculate the expression [tex]\( e^{256} \)[/tex], which we know is a very large number:
[tex]\[
e^{256} \approx 1.5114276650041035 \times 10^{111}
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Using the expression from step 2, we'll solve for [tex]\( P \)[/tex]:
[tex]\[
246.4 = P \cdot 1.5114276650041035 \times 10^{111}
\][/tex]
[tex]\[
P = \frac{246.4}{1.5114276650041035 \times 10^{111}}
\][/tex]
[tex]\[
P \approx 1.6302467243732173 \times 10^{-109}
\][/tex]
6. Interpret the Value of [tex]\( P \)[/tex]:
The computed value for [tex]\( P \approx 1.6302467243732173 \times 10^{-109} \)[/tex] is very small, and clearly too small to match any given multiple-choice option directly. Therefore, none of the options (A. 50, B. 210, C. 289, D. 1220) are approximate values for [tex]\( P \)[/tex].
Thus, based on calculations, the parameter [tex]\( P \)[/tex] in this scenario is much smaller than any of the given options.
1. Understand the Function and Given Values:
The function given is [tex]\( f(t) = P e^{t^t} \)[/tex]. We need to substitute the values [tex]\( t = 4 \)[/tex], and [tex]\( f(4) = 246.4 \)[/tex].
2. Substitute the Known Values in the Function:
By substituting [tex]\( t = 4 \)[/tex] into the function, we have:
[tex]\[
246.4 = P \cdot e^{4^4}
\][/tex]
3. Calculate [tex]\( 4^4 \)[/tex]:
First, compute the power:
[tex]\[
4^4 = 256
\][/tex]
4. Calculate [tex]\( e^{256} \)[/tex]:
Next, calculate the expression [tex]\( e^{256} \)[/tex], which we know is a very large number:
[tex]\[
e^{256} \approx 1.5114276650041035 \times 10^{111}
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Using the expression from step 2, we'll solve for [tex]\( P \)[/tex]:
[tex]\[
246.4 = P \cdot 1.5114276650041035 \times 10^{111}
\][/tex]
[tex]\[
P = \frac{246.4}{1.5114276650041035 \times 10^{111}}
\][/tex]
[tex]\[
P \approx 1.6302467243732173 \times 10^{-109}
\][/tex]
6. Interpret the Value of [tex]\( P \)[/tex]:
The computed value for [tex]\( P \approx 1.6302467243732173 \times 10^{-109} \)[/tex] is very small, and clearly too small to match any given multiple-choice option directly. Therefore, none of the options (A. 50, B. 210, C. 289, D. 1220) are approximate values for [tex]\( P \)[/tex].
Thus, based on calculations, the parameter [tex]\( P \)[/tex] in this scenario is much smaller than any of the given options.
