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Answer :
To find the approximate value of [tex]\( P \)[/tex] for the given function [tex]\( f(t) = P e^{r \cdot t} \)[/tex], we are provided with the condition [tex]\( f(4) = 246.4 \)[/tex] and the rate [tex]\( r = 0.04 \)[/tex].
Here are the steps to solve this problem:
1. Write down the given information:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
2. Substitute these values into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]
3. Solve the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{f(4)}{e^{0.04 \times 4}}
\][/tex]
4. Calculate the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]
5. Calculate [tex]\( e^{0.16} \)[/tex]:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
6. Substitute the values back to find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
So, the approximate value of [tex]\( P \)[/tex] is 210.
Therefore, the correct choice is [tex]\( \boxed{210} \)[/tex].
Here are the steps to solve this problem:
1. Write down the given information:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
2. Substitute these values into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]
3. Solve the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{f(4)}{e^{0.04 \times 4}}
\][/tex]
4. Calculate the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]
5. Calculate [tex]\( e^{0.16} \)[/tex]:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
6. Substitute the values back to find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
So, the approximate value of [tex]\( P \)[/tex] is 210.
Therefore, the correct choice is [tex]\( \boxed{210} \)[/tex].
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