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Answer :
To solve this problem, we need to find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], given that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].
Let's break it down:
1. Set Up the Equation:
- We know that [tex]\( f(4) = 246.4 \)[/tex].
- Substituting into the function, we have:
[tex]\[
246.4 = P \times e^{0.04 \times 4}
\][/tex]
2. Calculate the Exponential Part:
- Compute [tex]\( e^{0.16} \)[/tex], since [tex]\( 0.04 \times 4 = 0.16 \)[/tex].
3. Solve for [tex]\( P \)[/tex]:
- Divide both sides of the equation by [tex]\( e^{0.16} \)[/tex] to isolate [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
4. Calculate the Value:
- The result of calculating [tex]\( e^{0.16} \)[/tex] is approximately 1.1735.
- Now, divide 246.4 by this value:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
5. Select the Closest Option:
- The closest value to 209.97 from the given options is 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is C. 210.
Let's break it down:
1. Set Up the Equation:
- We know that [tex]\( f(4) = 246.4 \)[/tex].
- Substituting into the function, we have:
[tex]\[
246.4 = P \times e^{0.04 \times 4}
\][/tex]
2. Calculate the Exponential Part:
- Compute [tex]\( e^{0.16} \)[/tex], since [tex]\( 0.04 \times 4 = 0.16 \)[/tex].
3. Solve for [tex]\( P \)[/tex]:
- Divide both sides of the equation by [tex]\( e^{0.16} \)[/tex] to isolate [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
4. Calculate the Value:
- The result of calculating [tex]\( e^{0.16} \)[/tex] is approximately 1.1735.
- Now, divide 246.4 by this value:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
5. Select the Closest Option:
- The closest value to 209.97 from the given options is 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is C. 210.
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