Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex], given [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex], we can follow these steps:
1. Understand the function: We have the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. We know that when [tex]\( t = 3 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex].
2. Substitute known values: Use the given values:
[tex]\[
191.5 = P \cdot e^{0.03 \cdot 3}
\][/tex]
3. Calculate the exponent: Compute [tex]\( e^{0.03 \cdot 3} \)[/tex]. This calculates as:
[tex]\[
e^{0.09}
\][/tex]
4. Rearrange to solve for [tex]\( P \)[/tex]: Solve for [tex]\( P \)[/tex] by rearranging the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Compute the value of [tex]\( P \)[/tex]: Calculate the value, which comes out to approximately 175.018.
6. Choose the closest answer: Compare this result with the options available:
- A. 78
- B. 210
- C. 471
- D. 175
The closest answer to 175.018 is D. 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is 175.
1. Understand the function: We have the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. We know that when [tex]\( t = 3 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex].
2. Substitute known values: Use the given values:
[tex]\[
191.5 = P \cdot e^{0.03 \cdot 3}
\][/tex]
3. Calculate the exponent: Compute [tex]\( e^{0.03 \cdot 3} \)[/tex]. This calculates as:
[tex]\[
e^{0.09}
\][/tex]
4. Rearrange to solve for [tex]\( P \)[/tex]: Solve for [tex]\( P \)[/tex] by rearranging the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Compute the value of [tex]\( P \)[/tex]: Calculate the value, which comes out to approximately 175.018.
6. Choose the closest answer: Compare this result with the options available:
- A. 78
- B. 210
- C. 471
- D. 175
The closest answer to 175.018 is D. 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is 175.
