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Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P \cdot e^{-t} \)[/tex], where [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex], let's go through the steps.
1. Substitute [tex]\( t \)[/tex] and [tex]\( r \)[/tex] into the function:
The problem states [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex]. Substitute these values into the function:
[tex]\[
f(t) = P \cdot e^{-t \cdot r}
\][/tex]
Specifically for [tex]\( t = 3 \)[/tex]:
[tex]\[
f(3) = P \cdot e^{-3 \cdot 0.03}
\][/tex]
2. Simplify the exponent:
Calculate the exponent:
[tex]\[
e^{-3 \cdot 0.03} = e^{-0.09}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
Using the given [tex]\( f(3) = 191.5 \)[/tex], the equation becomes:
[tex]\[
191.5 = P \cdot e^{-0.09}
\][/tex]
Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( e^{-0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{-0.09}}
\][/tex]
4. Approximate [tex]\( e^{-0.09} \)[/tex]:
The value of [tex]\( e^{-0.09} \)[/tex] is approximately [tex]\( 0.9139 \)[/tex].
5. Calculate [tex]\( P \)[/tex]:
Substitute the approximate value of [tex]\( e^{-0.09} \)[/tex] back into the equation:
[tex]\[
P = \frac{191.5}{0.9139} \approx 209.53
\][/tex]
The approximate value of [tex]\( P \)[/tex] is 210.
Therefore, the answer is D. 210.
1. Substitute [tex]\( t \)[/tex] and [tex]\( r \)[/tex] into the function:
The problem states [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex]. Substitute these values into the function:
[tex]\[
f(t) = P \cdot e^{-t \cdot r}
\][/tex]
Specifically for [tex]\( t = 3 \)[/tex]:
[tex]\[
f(3) = P \cdot e^{-3 \cdot 0.03}
\][/tex]
2. Simplify the exponent:
Calculate the exponent:
[tex]\[
e^{-3 \cdot 0.03} = e^{-0.09}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
Using the given [tex]\( f(3) = 191.5 \)[/tex], the equation becomes:
[tex]\[
191.5 = P \cdot e^{-0.09}
\][/tex]
Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( e^{-0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{-0.09}}
\][/tex]
4. Approximate [tex]\( e^{-0.09} \)[/tex]:
The value of [tex]\( e^{-0.09} \)[/tex] is approximately [tex]\( 0.9139 \)[/tex].
5. Calculate [tex]\( P \)[/tex]:
Substitute the approximate value of [tex]\( e^{-0.09} \)[/tex] back into the equation:
[tex]\[
P = \frac{191.5}{0.9139} \approx 209.53
\][/tex]
The approximate value of [tex]\( P \)[/tex] is 210.
Therefore, the answer is D. 210.
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