Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], we are given that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
Here's how we can find [tex]\( P \)[/tex]:
1. We know that [tex]\( f(t) = P e^{rt} \)[/tex]. In this problem, [tex]\( t = 3 \)[/tex], so the function becomes:
[tex]\[
f(3) = P e^{3 \times 0.03}
\][/tex]
where [tex]\( e^{3 \times 0.03} = e^{0.09} \)[/tex].
2. We are given that [tex]\( f(3) = 191.5 \)[/tex]. So, we can set up the equation:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
3. To solve for [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
4. Calculate the value of [tex]\( e^{0.09} \)[/tex]. Approximating this step using the mathematical constant [tex]\( e \approx 2.71828 \)[/tex], you'll find:
[tex]\[
e^{0.09} \approx 1.09417
\][/tex]
5. Now, divide 191.5 by the value from step 4:
[tex]\[
P \approx \frac{191.5}{1.09417} \approx 175.0178
\][/tex]
6. The approximate value of [tex]\( P \)[/tex] corresponds to choice A, which is 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is 175.
Here's how we can find [tex]\( P \)[/tex]:
1. We know that [tex]\( f(t) = P e^{rt} \)[/tex]. In this problem, [tex]\( t = 3 \)[/tex], so the function becomes:
[tex]\[
f(3) = P e^{3 \times 0.03}
\][/tex]
where [tex]\( e^{3 \times 0.03} = e^{0.09} \)[/tex].
2. We are given that [tex]\( f(3) = 191.5 \)[/tex]. So, we can set up the equation:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
3. To solve for [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
4. Calculate the value of [tex]\( e^{0.09} \)[/tex]. Approximating this step using the mathematical constant [tex]\( e \approx 2.71828 \)[/tex], you'll find:
[tex]\[
e^{0.09} \approx 1.09417
\][/tex]
5. Now, divide 191.5 by the value from step 4:
[tex]\[
P \approx \frac{191.5}{1.09417} \approx 175.0178
\][/tex]
6. The approximate value of [tex]\( P \)[/tex] corresponds to choice A, which is 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is 175.
