Answer :
To find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex], we have the information that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
Let's solve it step by step:
1. Substitute the given values into the function:
You are given the function:
[tex]\[
f(t) = P \cdot e^{rt}
\][/tex]
We know:
[tex]\[
f(3) = 191.5
\][/tex]
[tex]\[
r = 0.03
\][/tex]
[tex]\[
t = 3
\][/tex]
Substituting these into the function, we get:
[tex]\[
191.5 = P \cdot e^{0.03 \cdot 3}
\][/tex]
2. Calculate [tex]\( e^{0.09} \)[/tex]:
The value of the exponent is [tex]\( 0.03 \times 3 = 0.09 \)[/tex]. Next, calculate:
[tex]\[
e^{0.09}
\][/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 [tex]\( P \)[/tex]:
After computing the value of [tex]\( e^{0.09} \)[/tex], divide 191.5 by that value to find [tex]\( P \)[/tex].
The approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the correct choice from the options provided is:
- A. 175
Let's solve it step by step:
1. Substitute the given values into the function:
You are given the function:
[tex]\[
f(t) = P \cdot e^{rt}
\][/tex]
We know:
[tex]\[
f(3) = 191.5
\][/tex]
[tex]\[
r = 0.03
\][/tex]
[tex]\[
t = 3
\][/tex]
Substituting these into the function, we get:
[tex]\[
191.5 = P \cdot e^{0.03 \cdot 3}
\][/tex]
2. Calculate [tex]\( e^{0.09} \)[/tex]:
The value of the exponent is [tex]\( 0.03 \times 3 = 0.09 \)[/tex]. Next, calculate:
[tex]\[
e^{0.09}
\][/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 [tex]\( P \)[/tex]:
After computing the value of [tex]\( e^{0.09} \)[/tex], divide 191.5 by that value to find [tex]\( P \)[/tex].
The approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the correct choice from the options provided is:
- A. 175
