Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^{0.04 \cdot t} \)[/tex], given that [tex]\( f(4) = 246.4 \)[/tex], we can follow these steps:
1. Understand the Given Function: The problem provides a function of the form [tex]\( f(t) = P \cdot e^{0.04 \cdot t} \)[/tex]. Here, [tex]\( P \)[/tex] is the initial value we need to determine, and [tex]\( e^{0.04 \cdot t} \)[/tex] is the exponential growth factor.
2. Substitute the Known Values: We know that at time [tex]\( t = 4 \)[/tex], the function value [tex]\( f(4) \)[/tex] is 246.4. This translates to the equation:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
3. Calculate the Exponential Part: First, calculate [tex]\( e^{0.04 \cdot 4} \)[/tex]:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
This is the approximate value of [tex]\( e^{0.16} \)[/tex].
4. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
Substitute the value we calculated for [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
5. Choose the Closest Option: Now, compare this result with the given options:
- A. 50
- B. 289
- C. 210
- D. 1220
The value 209.97 is closest to option C, which is 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{210}\)[/tex].
1. Understand the Given Function: The problem provides a function of the form [tex]\( f(t) = P \cdot e^{0.04 \cdot t} \)[/tex]. Here, [tex]\( P \)[/tex] is the initial value we need to determine, and [tex]\( e^{0.04 \cdot t} \)[/tex] is the exponential growth factor.
2. Substitute the Known Values: We know that at time [tex]\( t = 4 \)[/tex], the function value [tex]\( f(4) \)[/tex] is 246.4. This translates to the equation:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
3. Calculate the Exponential Part: First, calculate [tex]\( e^{0.04 \cdot 4} \)[/tex]:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
This is the approximate value of [tex]\( e^{0.16} \)[/tex].
4. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
Substitute the value we calculated for [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
5. Choose the Closest Option: Now, compare this result with the given options:
- A. 50
- B. 289
- C. 210
- D. 1220
The value 209.97 is closest to option C, which is 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{210}\)[/tex].
