Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], follow these steps:
1. Understand the Function:
The function is expressed as [tex]\( f(t) = P e^{rt} \)[/tex]. Here, [tex]\( P \)[/tex] is the initial value or principal amount, [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.718), and [tex]\( rt \)[/tex] is the product of the rate [tex]\( r \)[/tex] and time [tex]\( t \)[/tex].
2. Substitute the Given Values:
We know that [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex]. Substitute these values into the equation:
[tex]\[
246.4 = P \times e^{0.04 \times 4}
\][/tex]
3. Calculate the Exponent:
Calculate the value of the exponent [tex]\( rt = 0.04 \times 4 = 0.16 \)[/tex].
4. Evaluate [tex]\( e^{0.16} \)[/tex]:
Compute [tex]\( e^{0.16} \)[/tex], which is approximately equal to 1.1735.
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
6. Perform the Division:
Divide 246.4 by 1.1735 to find the approximate value of [tex]\( P \)[/tex]:
[tex]\[
P \approx 209.97
\][/tex]
7. Select the Closest Answer:
Compare the calculated value of [tex]\( P \)[/tex] to the provided options. The closest option to 209.97 is 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{210} \)[/tex].
1. Understand the Function:
The function is expressed as [tex]\( f(t) = P e^{rt} \)[/tex]. Here, [tex]\( P \)[/tex] is the initial value or principal amount, [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.718), and [tex]\( rt \)[/tex] is the product of the rate [tex]\( r \)[/tex] and time [tex]\( t \)[/tex].
2. Substitute the Given Values:
We know that [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex]. Substitute these values into the equation:
[tex]\[
246.4 = P \times e^{0.04 \times 4}
\][/tex]
3. Calculate the Exponent:
Calculate the value of the exponent [tex]\( rt = 0.04 \times 4 = 0.16 \)[/tex].
4. Evaluate [tex]\( e^{0.16} \)[/tex]:
Compute [tex]\( e^{0.16} \)[/tex], which is approximately equal to 1.1735.
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
6. Perform the Division:
Divide 246.4 by 1.1735 to find the approximate value of [tex]\( P \)[/tex]:
[tex]\[
P \approx 209.97
\][/tex]
7. Select the Closest Answer:
Compare the calculated value of [tex]\( P \)[/tex] to the provided options. The closest option to 209.97 is 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{210} \)[/tex].
