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Answer :
To solve for [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^{0.04t} \)[/tex] where [tex]\( f(4) = 246.4 \)[/tex], follow these steps:
1. We know the equation given is [tex]\( f(t) = P \cdot e^{0.04t} \)[/tex].
2. You're given that [tex]\( f(4) = 246.4 \)[/tex]. So, plug in [tex]\( t = 4 \)[/tex] into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]
3. Simplify the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]
4. Now the equation becomes:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
5. To solve for [tex]\( P \)[/tex], divide both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
6. Calculate [tex]\( e^{0.16} \)[/tex] which is approximately equal to 1.1735.
7. Finally, solve for [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210, which corresponds to option D.
1. We know the equation given is [tex]\( f(t) = P \cdot e^{0.04t} \)[/tex].
2. You're given that [tex]\( f(4) = 246.4 \)[/tex]. So, plug in [tex]\( t = 4 \)[/tex] into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]
3. Simplify the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]
4. Now the equation becomes:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
5. To solve for [tex]\( P \)[/tex], divide both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
6. Calculate [tex]\( e^{0.16} \)[/tex] which is approximately equal to 1.1735.
7. Finally, solve for [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210, which corresponds to option D.
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