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------------------------------------------------ If [tex]f(3) = 191.5[/tex] when [tex]r = 0.03[/tex] for the function [tex]f(t) = P e^{rt}[/tex], then what is the approximate value of [tex]P[/tex]?

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

Answer :

Sure, let's solve this problem step-by-step!

We are given a function [tex]\( f(t) = P e^{r \cdot t} \)[/tex]. We know that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex]. We need to find the approximate value of [tex]\( P \)[/tex].

1. Write down the function with given values:

[tex]\[
f(t) = P e^{r \cdot t}
\][/tex]

For [tex]\( t = 3 \)[/tex], the equation becomes:

[tex]\[
f(3) = P e^{0.03 \cdot 3}
\][/tex]

2. Use the given value of [tex]\( f(3) \)[/tex]:

[tex]\[
191.5 = P e^{0.03 \cdot 3}
\][/tex]

3. Calculate the exponent:

[tex]\[
0.03 \cdot 3 = 0.09
\][/tex]

So, our equation now is:

[tex]\[
191.5 = P e^{0.09}
\][/tex]

4. Solve for [tex]\( P \)[/tex]:

We need to isolate [tex]\( P \)[/tex], so we rearrange the equation:

[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]

5. Approximate [tex]\( e^{0.09} \)[/tex]:

Calculating [tex]\( e^{0.09} \)[/tex], we find it's approximately equal to 1.0942.

6. Calculate [tex]\( P \)[/tex]:

[tex]\[
P \approx \frac{191.5}{1.0942} \approx 175.0178
\][/tex]

7. Select the closest option:

The value of [tex]\( P \)[/tex] is approximately 175, which matches option B.

So, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].