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Answer :
To find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^t \)[/tex] given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], we need to follow these steps:
1. Understand the Function: The function is given in the form [tex]\( f(t) = P e^t \)[/tex].
2. Substitute Known Values: We know that [tex]\( f(4) = 246.4 \)[/tex] and we need to find [tex]\( P \)[/tex] such that:
[tex]\[
246.4 = P e^4
\][/tex]
3. Solve for [tex]\( P \)[/tex]: Rearrange the equation to get [tex]\( P \)[/tex] by itself:
[tex]\[
P = \frac{246.4}{e^4}
\][/tex]
4. Calculate [tex]\( e^4 \)[/tex]: Use the value of the exponential function [tex]\( e \)[/tex]. A precise value for [tex]\( e^4 \)[/tex] is approximately 54.598.
5. Calculate [tex]\( P \)[/tex]: Divide 246.4 by the value of [tex]\( e^4 \)[/tex]:
[tex]\[
P \approx \frac{246.4}{54.598} \approx 4.51
\][/tex]
Given these calculations, [tex]\( P \approx 4.51 \)[/tex], which is not directly listed among the provided options (A. 289, B. 210, C. 1220, D. 50). Therefore, none of the options match the calculated result; however, this means an error may exist in the options provided or they may not be related to the correct context of [tex]\( P \)[/tex] as directly calculated here.
1. Understand the Function: The function is given in the form [tex]\( f(t) = P e^t \)[/tex].
2. Substitute Known Values: We know that [tex]\( f(4) = 246.4 \)[/tex] and we need to find [tex]\( P \)[/tex] such that:
[tex]\[
246.4 = P e^4
\][/tex]
3. Solve for [tex]\( P \)[/tex]: Rearrange the equation to get [tex]\( P \)[/tex] by itself:
[tex]\[
P = \frac{246.4}{e^4}
\][/tex]
4. Calculate [tex]\( e^4 \)[/tex]: Use the value of the exponential function [tex]\( e \)[/tex]. A precise value for [tex]\( e^4 \)[/tex] is approximately 54.598.
5. Calculate [tex]\( P \)[/tex]: Divide 246.4 by the value of [tex]\( e^4 \)[/tex]:
[tex]\[
P \approx \frac{246.4}{54.598} \approx 4.51
\][/tex]
Given these calculations, [tex]\( P \approx 4.51 \)[/tex], which is not directly listed among the provided options (A. 289, B. 210, C. 1220, D. 50). Therefore, none of the options match the calculated result; however, this means an error may exist in the options provided or they may not be related to the correct context of [tex]\( P \)[/tex] as directly calculated here.
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