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Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex], we can follow these steps:
1. Understand the Function: We have the function [tex]\( f(t) = P e^{rt} \)[/tex]. We know that for [tex]\( t = 4 \)[/tex], the function's value is [tex]\( f(4) = 246.4 \)[/tex].
2. Plug in the Known Values: We need to substitute the given values into the function:
[tex]\[
246.4 = P e^{0.04 \times 4}
\][/tex]
3. Calculate the Exponent: Calculate [tex]\( e^{0.04 \times 4} \)[/tex]. The exponent calculation gives us approximately:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
4. Solve for [tex]\( P \)[/tex]: Use the equation obtained from plugging in the values and divide both sides by [tex]\( e^{0.16} \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
5. Determine the Closest Answer: The calculated value of [tex]\( P \)[/tex] is approximately 209.97. Among the given options, the one closest to this value is:
[tex]\[
\text{A. } 210
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{210}\)[/tex].
1. Understand the Function: We have the function [tex]\( f(t) = P e^{rt} \)[/tex]. We know that for [tex]\( t = 4 \)[/tex], the function's value is [tex]\( f(4) = 246.4 \)[/tex].
2. Plug in the Known Values: We need to substitute the given values into the function:
[tex]\[
246.4 = P e^{0.04 \times 4}
\][/tex]
3. Calculate the Exponent: Calculate [tex]\( e^{0.04 \times 4} \)[/tex]. The exponent calculation gives us approximately:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
4. Solve for [tex]\( P \)[/tex]: Use the equation obtained from plugging in the values and divide both sides by [tex]\( e^{0.16} \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
5. Determine the Closest Answer: The calculated value of [tex]\( P \)[/tex] is approximately 209.97. Among the given options, the one closest to this value is:
[tex]\[
\text{A. } 210
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{210}\)[/tex].
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