Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex] given that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex], we can follow these steps:
1. Understand the Function:
The function is given as [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. This implies that the value of the function for a specific time [tex]\( t \)[/tex] is determined by the initial value [tex]\( P \)[/tex] and the growth factor [tex]\( e^{rt} \)[/tex].
2. Substitute the Known Values:
We need to substitute the known values into the function. We have:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
3. Setting Up the Equation:
Substitute these into the equation:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
4. Calculate [tex]\( e^{0.04 \cdot 4} \)[/tex]:
Calculate the exponent part:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
This is the value of the growth factor when you multiply the rate and time.
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
6. Calculate the Value of [tex]\( P \)[/tex]:
When you calculate [tex]\( \frac{246.4}{1.1735} \)[/tex], you get approximately:
[tex]\[
P \approx 210
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210. The correct choice is C. 210.
1. Understand the Function:
The function is given as [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. This implies that the value of the function for a specific time [tex]\( t \)[/tex] is determined by the initial value [tex]\( P \)[/tex] and the growth factor [tex]\( e^{rt} \)[/tex].
2. Substitute the Known Values:
We need to substitute the known values into the function. We have:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
3. Setting Up the Equation:
Substitute these into the equation:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
4. Calculate [tex]\( e^{0.04 \cdot 4} \)[/tex]:
Calculate the exponent part:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
This is the value of the growth factor when you multiply the rate and time.
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
6. Calculate the Value of [tex]\( P \)[/tex]:
When you calculate [tex]\( \frac{246.4}{1.1735} \)[/tex], you get approximately:
[tex]\[
P \approx 210
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210. The correct choice is C. 210.
