Answer :
To find the approximate value of [tex]\( P \)[/tex], we start from the given function:
[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]
We're informed that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex]. Let's set up the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
Calculate the exponent:
[tex]\[ e^{0.03 \cdot 3} = e^{0.09} \][/tex]
To find [tex]\( P \)[/tex], rearrange the equation:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
Substitute the value of [tex]\( e^{0.09} \)[/tex]:
- Calculating [tex]\( e^{0.09} \)[/tex], you get approximately 1.09417.
Now, compute [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{1.09417} \][/tex]
[tex]\[ P \approx 175.0178 \][/tex]
The approximate value of [tex]\( P \)[/tex] is about 175. Among the given choices:
- A. 471
- B. 210
- C. 175
- D. 78
The closest to the calculated value is:
C. 175
Therefore, the approximate value of [tex]\( P \)[/tex] is 175.
[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]
We're informed that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex]. Let's set up the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
Calculate the exponent:
[tex]\[ e^{0.03 \cdot 3} = e^{0.09} \][/tex]
To find [tex]\( P \)[/tex], rearrange the equation:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
Substitute the value of [tex]\( e^{0.09} \)[/tex]:
- Calculating [tex]\( e^{0.09} \)[/tex], you get approximately 1.09417.
Now, compute [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{1.09417} \][/tex]
[tex]\[ P \approx 175.0178 \][/tex]
The approximate value of [tex]\( P \)[/tex] is about 175. Among the given choices:
- A. 471
- B. 210
- C. 175
- D. 78
The closest to the calculated value is:
C. 175
Therefore, the approximate value of [tex]\( P \)[/tex] is 175.
