Answer :
To solve the problem, we want to find the initial value [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
Here's how you can solve it step-by-step:
1. Substitute the Known Values:
- We know [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
- The equation becomes [tex]\( f(3) = P e^{0.03 \times 3} = 191.5 \)[/tex].
2. Calculate the Exponential Term:
- Compute [tex]\( e^{0.03 \times 3} = e^{0.09} \)[/tex]. This gives a numerical value.
3. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
- Calculate the value of [tex]\( e^{0.09} \)[/tex] and then divide 191.5 by this value to find [tex]\( P \)[/tex].
4. Determine [tex]\( P \)[/tex]:
- The proximate value obtained for [tex]\( P \)[/tex] is around 175.
With these calculations, the closest option that matches the result is:
- B. 175
Here's how you can solve it step-by-step:
1. Substitute the Known Values:
- We know [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
- The equation becomes [tex]\( f(3) = P e^{0.03 \times 3} = 191.5 \)[/tex].
2. Calculate the Exponential Term:
- Compute [tex]\( e^{0.03 \times 3} = e^{0.09} \)[/tex]. This gives a numerical value.
3. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
- Calculate the value of [tex]\( e^{0.09} \)[/tex] and then divide 191.5 by this value to find [tex]\( P \)[/tex].
4. Determine [tex]\( P \)[/tex]:
- The proximate value obtained for [tex]\( P \)[/tex] is around 175.
With these calculations, the closest option that matches the result is:
- B. 175
