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Answer :
To find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^t \)[/tex] with the information given, we'll follow these steps:
1. We know that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
2. The function is given as [tex]\( f(t) = P e^{rt} \)[/tex].
Let's plug in the known values:
- Since [tex]\( f(3) = 191.5 \)[/tex], we substitute 3 for [tex]\( t \)[/tex]:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
3. Next, simplify the exponent:
[tex]\[
e^{0.03 \times 3} = e^{0.09}
\][/tex]
4. To solve for [tex]\( P \)[/tex], rearrange the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Now, calculate the approximate value of [tex]\( e^{0.09} \)[/tex] and divide 191.5 by this result to find [tex]\( P \)[/tex].
6. After calculating, the approximate value of [tex]\( P \)[/tex] is found to be 175.
Therefore, the correct option is:
C. 175
1. We know that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
2. The function is given as [tex]\( f(t) = P e^{rt} \)[/tex].
Let's plug in the known values:
- Since [tex]\( f(3) = 191.5 \)[/tex], we substitute 3 for [tex]\( t \)[/tex]:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
3. Next, simplify the exponent:
[tex]\[
e^{0.03 \times 3} = e^{0.09}
\][/tex]
4. To solve for [tex]\( P \)[/tex], rearrange the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Now, calculate the approximate value of [tex]\( e^{0.09} \)[/tex] and divide 191.5 by this result to find [tex]\( P \)[/tex].
6. After calculating, the approximate value of [tex]\( P \)[/tex] is found to be 175.
Therefore, the correct option is:
C. 175
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