Answer :
To solve for the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], where [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex], we can follow these steps:
1. Start with the given function:
[tex]\( f(t) = P e^{rt} \)[/tex].
2. Substitute the known values into the function:
We know that [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex]. So,
[tex]\( 191.5 = P e^{0.03 \times 3} \)[/tex].
3. Calculate the exponent:
First, calculate [tex]\( 0.03 \times 3 = 0.09 \)[/tex].
4. Evaluate the exponential term:
Find [tex]\( e^{0.09} \)[/tex]. This is approximately equal to [tex]\( 1.094 \)[/tex] (rounded to three decimal places).
5. Solve for [tex]\( P \)[/tex]:
Replace [tex]\( e^{0.09} \)[/tex] with its approximate value:
[tex]\( 191.5 = P \times 1.094 \)[/tex].
6. Isolate [tex]\( P \)[/tex]:
Divide both sides by [tex]\( 1.094 \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\( P = \frac{191.5}{1.094} \)[/tex].
7. Calculate [tex]\( P \)[/tex]:
[tex]\( P \approx 175.02 \)[/tex].
The approximate value of [tex]\( P \)[/tex] is 175. Therefore, the closest option is:
D. 175
1. Start with the given function:
[tex]\( f(t) = P e^{rt} \)[/tex].
2. Substitute the known values into the function:
We know that [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex]. So,
[tex]\( 191.5 = P e^{0.03 \times 3} \)[/tex].
3. Calculate the exponent:
First, calculate [tex]\( 0.03 \times 3 = 0.09 \)[/tex].
4. Evaluate the exponential term:
Find [tex]\( e^{0.09} \)[/tex]. This is approximately equal to [tex]\( 1.094 \)[/tex] (rounded to three decimal places).
5. Solve for [tex]\( P \)[/tex]:
Replace [tex]\( e^{0.09} \)[/tex] with its approximate value:
[tex]\( 191.5 = P \times 1.094 \)[/tex].
6. Isolate [tex]\( P \)[/tex]:
Divide both sides by [tex]\( 1.094 \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\( P = \frac{191.5}{1.094} \)[/tex].
7. Calculate [tex]\( P \)[/tex]:
[tex]\( P \approx 175.02 \)[/tex].
The approximate value of [tex]\( P \)[/tex] is 175. Therefore, the closest option is:
D. 175
