Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] using the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].
Here's a step-by-step solution:
1. Substitute known values into the equation:
We start with the equation [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. Substitute [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex]:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
2. Calculate [tex]\( e^{0.04 \cdot 4} \)[/tex]:
Compute the exponent:
[tex]\[
0.04 \cdot 4 = 0.16
\][/tex]
So, [tex]\( e^{0.16} \)[/tex] is approximately 1.17351087099.
3. Solve for [tex]\( P \)[/tex]:
Now, divide 246.4 by the computed value of [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351087099} \approx 209.97
\][/tex]
4. Select the closest value from the options:
The result is approximately 209.97. Looking at the options:
- A. 289
- B. 1220
- C. 210
- D. 50
The value closest to 209.97 is 210. Thus, the approximate value of [tex]\( P \)[/tex] is:
[tex]\[
\boxed{210}
\][/tex]
Here's a step-by-step solution:
1. Substitute known values into the equation:
We start with the equation [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. Substitute [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex]:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
2. Calculate [tex]\( e^{0.04 \cdot 4} \)[/tex]:
Compute the exponent:
[tex]\[
0.04 \cdot 4 = 0.16
\][/tex]
So, [tex]\( e^{0.16} \)[/tex] is approximately 1.17351087099.
3. Solve for [tex]\( P \)[/tex]:
Now, divide 246.4 by the computed value of [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351087099} \approx 209.97
\][/tex]
4. Select the closest value from the options:
The result is approximately 209.97. Looking at the options:
- A. 289
- B. 1220
- C. 210
- D. 50
The value closest to 209.97 is 210. Thus, the approximate value of [tex]\( P \)[/tex] is:
[tex]\[
\boxed{210}
\][/tex]
