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Answer :
To find the approximate value of [tex]\( P \)[/tex] for the given function [tex]\( f(t) = P e^{rt} \)[/tex], where it's given that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex], follow these steps:
1. Identify the given values and the function:
- You know [tex]\( f(3) = 191.5 \)[/tex].
- The rate [tex]\( r = 0.03 \)[/tex].
- The function is [tex]\( f(t) = P e^{rt} \)[/tex].
2. Substitute the given values into the function:
- Substitute [tex]\( t = 3 \)[/tex] into the function: [tex]\( f(3) = P e^{0.03 \cdot 3} \)[/tex].
3. Plug in the known value of [tex]\( f(3) \)[/tex]:
- You know [tex]\( f(3) = 191.5 \)[/tex], so set up the equation:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
- To isolate [tex]\( P \)[/tex], divide both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Calculate the numerical value:
- The exponential term [tex]\( e^{0.09} \)[/tex] is a constant that you'll evaluate.
- When you compute this, you'll find that [tex]\( P \)[/tex] is approximately 175.
6. Compare with the options:
- The calculated value is closest to option A, which is 175.
So, the approximate value of [tex]\( P \)[/tex] is 175, which corresponds to option A.
1. Identify the given values and the function:
- You know [tex]\( f(3) = 191.5 \)[/tex].
- The rate [tex]\( r = 0.03 \)[/tex].
- The function is [tex]\( f(t) = P e^{rt} \)[/tex].
2. Substitute the given values into the function:
- Substitute [tex]\( t = 3 \)[/tex] into the function: [tex]\( f(3) = P e^{0.03 \cdot 3} \)[/tex].
3. Plug in the known value of [tex]\( f(3) \)[/tex]:
- You know [tex]\( f(3) = 191.5 \)[/tex], so set up the equation:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
- To isolate [tex]\( P \)[/tex], divide both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Calculate the numerical value:
- The exponential term [tex]\( e^{0.09} \)[/tex] is a constant that you'll evaluate.
- When you compute this, you'll find that [tex]\( P \)[/tex] is approximately 175.
6. Compare with the options:
- The calculated value is closest to option A, which is 175.
So, the approximate value of [tex]\( P \)[/tex] is 175, which corresponds to option A.
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