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Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. We know that:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
Substitute these values into the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
To find [tex]\( P \)[/tex], we first need to calculate the exponent:
1. Calculate the exponent [tex]\( 0.03 \cdot 3 \)[/tex], which equals [tex]\( 0.09 \)[/tex].
2. Evaluate [tex]\( e^{0.09} \)[/tex]. This equals approximately [tex]\( 1.09417 \)[/tex].
Now, we substitute back into the equation:
[tex]\[ 191.5 = P \cdot 1.09417 \][/tex]
Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( 1.09417 \)[/tex]:
[tex]\[ P = \frac{191.5}{1.09417} \][/tex]
This calculation yields [tex]\( P \approx 175.02 \)[/tex]. Thus, the approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the correct answer is:
D. 175
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
Substitute these values into the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
To find [tex]\( P \)[/tex], we first need to calculate the exponent:
1. Calculate the exponent [tex]\( 0.03 \cdot 3 \)[/tex], which equals [tex]\( 0.09 \)[/tex].
2. Evaluate [tex]\( e^{0.09} \)[/tex]. This equals approximately [tex]\( 1.09417 \)[/tex].
Now, we substitute back into the equation:
[tex]\[ 191.5 = P \cdot 1.09417 \][/tex]
Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( 1.09417 \)[/tex]:
[tex]\[ P = \frac{191.5}{1.09417} \][/tex]
This calculation yields [tex]\( P \approx 175.02 \)[/tex]. Thus, the approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the correct answer is:
D. 175
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