High School

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------------------------------------------------ For the function [tex]f(t) = P e^t[/tex], it is given that [tex]f(3) = 191.5[/tex] when [tex]r = 0.03[/tex]. What is the approximate value of [tex]P[/tex]?

A. 78
B. 175
C. 210
D. 471

Answer :

To solve the problem, we want to find the initial value [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].

Here's how you can solve it step-by-step:

1. Substitute the Known Values:
- We know [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
- The equation becomes [tex]\( f(3) = P e^{0.03 \times 3} = 191.5 \)[/tex].

2. Calculate the Exponential Term:
- Compute [tex]\( e^{0.03 \times 3} = e^{0.09} \)[/tex]. This gives a numerical value.

3. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
- Calculate the value of [tex]\( e^{0.09} \)[/tex] and then divide 191.5 by this value to find [tex]\( P \)[/tex].

4. Determine [tex]\( P \)[/tex]:
- The proximate value obtained for [tex]\( P \)[/tex] is around 175.

With these calculations, the closest option that matches the result is:
- B. 175