Answer :
To solve for the approximate value of [tex]\( P \)[/tex] in the equation [tex]\( f(t) = P \cdot e^{rt} \)[/tex], we are given the following information:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
The function can be rewritten as:
[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]
We can substitute the given values [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex] into the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
First, calculate the exponential part:
[tex]\[ e^{0.03 \cdot 3} = e^{0.09} \][/tex]
Now compute [tex]\( e^{0.09} \)[/tex]:
The approximate value is [tex]\( e^{0.09} \approx 1.094 \)[/tex].
Substitute this back into the equation:
[tex]\[ 191.5 = P \cdot 1.094 \][/tex]
Solve for [tex]\( P \)[/tex] by dividing both sides by 1.094:
[tex]\[ P \approx \frac{191.5}{1.094} \][/tex]
[tex]\[ P \approx 175 \][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 175, which corresponds to option B.
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
The function can be rewritten as:
[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]
We can substitute the given values [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex] into the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
First, calculate the exponential part:
[tex]\[ e^{0.03 \cdot 3} = e^{0.09} \][/tex]
Now compute [tex]\( e^{0.09} \)[/tex]:
The approximate value is [tex]\( e^{0.09} \approx 1.094 \)[/tex].
Substitute this back into the equation:
[tex]\[ 191.5 = P \cdot 1.094 \][/tex]
Solve for [tex]\( P \)[/tex] by dividing both sides by 1.094:
[tex]\[ P \approx \frac{191.5}{1.094} \][/tex]
[tex]\[ P \approx 175 \][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 175, which corresponds to option B.
