Answer :
Let's solve the problem step by step to find the approximate value of [tex]\( P \)[/tex].
We are given the function:
[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]
and the information:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
We need to find the value of [tex]\( P \)[/tex].
1. Substitute the given values into the function:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
2. Calculate [tex]\( e^{0.03 \cdot 3} \)[/tex]:
- [tex]\( 0.03 \times 3 = 0.09 \)[/tex]
- So, we need [tex]\( e^{0.09} \)[/tex], which is approximately [tex]\( 1.0942 \)[/tex].
3. Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
4. Substitute the approximate value [tex]\( e^{0.09} \approx 1.0942 \)[/tex] into the equation:
[tex]\[ P \approx \frac{191.5}{1.0942} \][/tex]
5. Perform the division:
[tex]\[ P \approx 175.02 \][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( 175 \)[/tex], which corresponds to option C.
We are given the function:
[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]
and the information:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
We need to find the value of [tex]\( P \)[/tex].
1. Substitute the given values into the function:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
2. Calculate [tex]\( e^{0.03 \cdot 3} \)[/tex]:
- [tex]\( 0.03 \times 3 = 0.09 \)[/tex]
- So, we need [tex]\( e^{0.09} \)[/tex], which is approximately [tex]\( 1.0942 \)[/tex].
3. Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
4. Substitute the approximate value [tex]\( e^{0.09} \approx 1.0942 \)[/tex] into the equation:
[tex]\[ P \approx \frac{191.5}{1.0942} \][/tex]
5. Perform the division:
[tex]\[ P \approx 175.02 \][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( 175 \)[/tex], which corresponds to option C.
