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Answer :
To find the approximate value of [tex]\( P \)[/tex], let's follow these steps based on the information given:
1. Understand the Problem:
We are given a function [tex]\( f(t) = P e^{t} \)[/tex] and specified that when [tex]\( r = 0.03 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex].
2. Express the Equation Using the Given Information:
Since [tex]\( f(t) = P \cdot e^{t} \)[/tex] and we know that [tex]\( f(3) = 191.5 \)[/tex], we can write:
[tex]\[
191.5 = P \cdot e^{3r}
\][/tex]
Here [tex]\( r = 0.03 \)[/tex].
3. Substitute the Known Values:
Substitute [tex]\( r = 0.03 \)[/tex] into the equation:
[tex]\[
191.5 = P \cdot e^{3 \times 0.03}
\][/tex]
Simplify to get:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
To find [tex]\( P \)[/tex], we need to isolate it by dividing both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Calculate the Value:
[tex]\(\ e^{0.09} \)[/tex] is approximately 1.094. Divide [tex]\( 191.5 \)[/tex] by 1.094 to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{191.5}{1.094} \approx 175.02
\][/tex]
6. Determine the Closest Answer:
Round 175.02 to the nearest whole number. The approximate value of [tex]\( P \)[/tex] is 175.
So, the correct answer is:
A. 175
1. Understand the Problem:
We are given a function [tex]\( f(t) = P e^{t} \)[/tex] and specified that when [tex]\( r = 0.03 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex].
2. Express the Equation Using the Given Information:
Since [tex]\( f(t) = P \cdot e^{t} \)[/tex] and we know that [tex]\( f(3) = 191.5 \)[/tex], we can write:
[tex]\[
191.5 = P \cdot e^{3r}
\][/tex]
Here [tex]\( r = 0.03 \)[/tex].
3. Substitute the Known Values:
Substitute [tex]\( r = 0.03 \)[/tex] into the equation:
[tex]\[
191.5 = P \cdot e^{3 \times 0.03}
\][/tex]
Simplify to get:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
To find [tex]\( P \)[/tex], we need to isolate it by dividing both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Calculate the Value:
[tex]\(\ e^{0.09} \)[/tex] is approximately 1.094. Divide [tex]\( 191.5 \)[/tex] by 1.094 to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{191.5}{1.094} \approx 175.02
\][/tex]
6. Determine the Closest Answer:
Round 175.02 to the nearest whole number. The approximate value of [tex]\( P \)[/tex] is 175.
So, the correct answer is:
A. 175
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