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Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], we follow these steps:
1. Identify the values given in the problem:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
2. Write the equation using the function definition:
[tex]\[
f(t) = P \cdot e^{r \cdot t}
\][/tex]
Substituting the given values, we have:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
3. Calculate the exponent in the exponential function:
[tex]\[
r \cdot t = 0.04 \cdot 4 = 0.16
\][/tex]
4. Calculate [tex]\( e^{0.16} \)[/tex]. The numerical approximation for [tex]\( e^{0.16} \)[/tex] is approximately 1.1735.
5. Solve for [tex]\( P \)[/tex] by rearranging the equation:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
6. Perform the division:
[tex]\[
P \approx 209.97
\][/tex]
The closest approximate value from the options given is 210. Therefore, the answer is:
C. 210
1. Identify the values given in the problem:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
2. Write the equation using the function definition:
[tex]\[
f(t) = P \cdot e^{r \cdot t}
\][/tex]
Substituting the given values, we have:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
3. Calculate the exponent in the exponential function:
[tex]\[
r \cdot t = 0.04 \cdot 4 = 0.16
\][/tex]
4. Calculate [tex]\( e^{0.16} \)[/tex]. The numerical approximation for [tex]\( e^{0.16} \)[/tex] is approximately 1.1735.
5. Solve for [tex]\( P \)[/tex] by rearranging the equation:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
6. Perform the division:
[tex]\[
P \approx 209.97
\][/tex]
The closest approximate value from the options given is 210. Therefore, the answer is:
C. 210
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