College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ If [tex]f(5)=288.9[/tex] when [tex]r=0.05[/tex] for the function [tex]f(t)=P e^{rt}[/tex], then what is the approximate value of [tex]P[/tex]?

A. 3520
B. 371
C. 24
D. 225

Answer :

To find the approximate value of [tex]\( P \)[/tex], given the equation [tex]\( f(t) = P \cdot e^{rt} \)[/tex] where [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we can follow these steps:

1. Identify the components: We know [tex]\( f(5) = 288.9 \)[/tex], [tex]\( r = 0.05 \)[/tex], and we assume [tex]\( t = 5 \)[/tex].

2. Substitute known values into the function: Substitute the known values into the equation [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. So, it becomes:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]

3. Calculate [tex]\( e^{0.05 \times 5} \)[/tex]: First, compute the exponent:
[tex]\[
e^{0.25} \approx 1.284025
\][/tex]

4. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284025}
\][/tex]

5. Perform the division:
[tex]\[
P \approx 224.99554622932885
\][/tex]

6. Approximate [tex]\( P \)[/tex] to the nearest answer choice: Comparing the computed value of [tex]\( P \)[/tex] to the given options, it is closest to:
- [tex]\( \text{D. } 225 \)[/tex]

Thus, the approximate value of [tex]\( P \)[/tex] is 225.