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Answer :
Let's solve the problem step-by-step:
We have the function [tex]\( f(t) = P e^{r \cdot t} \)[/tex], and we're given that when [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex]. We need to find the approximate value of [tex]\( P \)[/tex].
Here's how we can find [tex]\( P \)[/tex]:
1. Substitute the known values into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
2. Solve for [tex]\( P \)[/tex]:
We need to calculate [tex]\( e^{0.16} \)[/tex]. This is an exponential function where [tex]\( e \approx 2.71828 \)[/tex]. When calculated, [tex]\( e^{0.16} \approx 1.17351 \)[/tex].
3. Divide [tex]\( f(4) \)[/tex] by [tex]\( e^{0.16} \)[/tex] to find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351} \approx 209.97
\][/tex]
4. Approximate [tex]\( P \)[/tex]:
The approximate value of [tex]\( P \)[/tex] is [tex]\( 210 \)[/tex].
Therefore, the correct option that closely matches this value is C. 210.
We have the function [tex]\( f(t) = P e^{r \cdot t} \)[/tex], and we're given that when [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex]. We need to find the approximate value of [tex]\( P \)[/tex].
Here's how we can find [tex]\( P \)[/tex]:
1. Substitute the known values into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
2. Solve for [tex]\( P \)[/tex]:
We need to calculate [tex]\( e^{0.16} \)[/tex]. This is an exponential function where [tex]\( e \approx 2.71828 \)[/tex]. When calculated, [tex]\( e^{0.16} \approx 1.17351 \)[/tex].
3. Divide [tex]\( f(4) \)[/tex] by [tex]\( e^{0.16} \)[/tex] to find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351} \approx 209.97
\][/tex]
4. Approximate [tex]\( P \)[/tex]:
The approximate value of [tex]\( P \)[/tex] is [tex]\( 210 \)[/tex].
Therefore, the correct option that closely matches this value is C. 210.
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