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Answer :
To find the approximate value of [tex]\( P \)[/tex], we need to use the given information and the formula for the function [tex]\( f(t) = P e^{rt} \)[/tex]. We are told that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
1. Start with the equation:
[tex]\[ f(t) = P e^{rt} \][/tex]
Here, [tex]\( t = 3 \)[/tex], so the function becomes:
[tex]\[ f(3) = P e^{0.03 \times 3} \][/tex]
2. Substitute the given value of [tex]\( f(3) \)[/tex], which is 191.5:
[tex]\[ 191.5 = P e^{0.09} \][/tex]
3. To solve for [tex]\( P \)[/tex], we need to isolate it:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
4. Calculate [tex]\( e^{0.09} \)[/tex]. Knowing that [tex]\( e^{0.09} \)[/tex] slightly above 1 gives a better sense of the division outcome, but for now, let's acknowledge the calculation has been performed:
5. Use this result to find [tex]\( P \)[/tex]:
[tex]\[ P \approx \frac{191.5}{e^{0.09}} \approx 175 \][/tex]
So, the approximate value of [tex]\( P \)[/tex] is 175, which corresponds to option B.
1. Start with the equation:
[tex]\[ f(t) = P e^{rt} \][/tex]
Here, [tex]\( t = 3 \)[/tex], so the function becomes:
[tex]\[ f(3) = P e^{0.03 \times 3} \][/tex]
2. Substitute the given value of [tex]\( f(3) \)[/tex], which is 191.5:
[tex]\[ 191.5 = P e^{0.09} \][/tex]
3. To solve for [tex]\( P \)[/tex], we need to isolate it:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
4. Calculate [tex]\( e^{0.09} \)[/tex]. Knowing that [tex]\( e^{0.09} \)[/tex] slightly above 1 gives a better sense of the division outcome, but for now, let's acknowledge the calculation has been performed:
5. Use this result to find [tex]\( P \)[/tex]:
[tex]\[ P \approx \frac{191.5}{e^{0.09}} \approx 175 \][/tex]
So, the approximate value of [tex]\( P \)[/tex] is 175, which corresponds to option B.
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