Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], we are given:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]
We can rearrange the formula of the function to solve for [tex]\( P \)[/tex]:
1. Start with the equation:
[tex]\[
f(t) = P e^{rt}
\][/tex]
2. Plug in the given values:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
3. Calculate [tex]\( e^{0.03 \times 3} \)[/tex]:
- First, calculate the exponent: [tex]\( 0.03 \times 3 = 0.09 \)[/tex]
- Then find [tex]\( e^{0.09} \)[/tex], which is approximately [tex]\( 1.09417 \)[/tex].
4. Now substitute back into the equation:
[tex]\[
191.5 = P \times 1.09417
\][/tex]
5. Solve for [tex]\( P \)[/tex] by dividing both sides by 1.09417:
[tex]\[
P = \frac{191.5}{1.09417}
\][/tex]
6. Calculate the result:
[tex]\[
P \approx 175.0178
\][/tex]
The approximate value of [tex]\( P \)[/tex] is therefore closest to option B, 175.
Hence, the correct answer is B. 175.
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]
We can rearrange the formula of the function to solve for [tex]\( P \)[/tex]:
1. Start with the equation:
[tex]\[
f(t) = P e^{rt}
\][/tex]
2. Plug in the given values:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
3. Calculate [tex]\( e^{0.03 \times 3} \)[/tex]:
- First, calculate the exponent: [tex]\( 0.03 \times 3 = 0.09 \)[/tex]
- Then find [tex]\( e^{0.09} \)[/tex], which is approximately [tex]\( 1.09417 \)[/tex].
4. Now substitute back into the equation:
[tex]\[
191.5 = P \times 1.09417
\][/tex]
5. Solve for [tex]\( P \)[/tex] by dividing both sides by 1.09417:
[tex]\[
P = \frac{191.5}{1.09417}
\][/tex]
6. Calculate the result:
[tex]\[
P \approx 175.0178
\][/tex]
The approximate value of [tex]\( P \)[/tex] is therefore closest to option B, 175.
Hence, the correct answer is B. 175.
