Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
Here's a step-by-step solution:
1. Understand the equation:
The function is given as [tex]\( f(t) = P e^{rt} \)[/tex].
We know [tex]\( f(3) = 191.5 \)[/tex], which means when [tex]\( t = 3 \)[/tex], the function value is 191.5.
2. Substitute known values into the equation:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
Simplify the exponent:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
4. Calculate the value:
Determine the numerical value of [tex]\( e^{0.09} \)[/tex] and divide 191.5 by this number to find [tex]\( P \)[/tex].
5. Result:
After calculation, the approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the answer closest to this calculated value is option D. 175.
Here's a step-by-step solution:
1. Understand the equation:
The function is given as [tex]\( f(t) = P e^{rt} \)[/tex].
We know [tex]\( f(3) = 191.5 \)[/tex], which means when [tex]\( t = 3 \)[/tex], the function value is 191.5.
2. Substitute known values into the equation:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
Simplify the exponent:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
4. Calculate the value:
Determine the numerical value of [tex]\( e^{0.09} \)[/tex] and divide 191.5 by this number to find [tex]\( P \)[/tex].
5. Result:
After calculation, the approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the answer closest to this calculated value is option D. 175.
