Answer :
Sure! Let's find the approximate value of [tex]\( P \)[/tex] given the function [tex]\( f(t) = Pe^{rt} \)[/tex] and the information [tex]\( f(4) = 246.4 \)[/tex] with [tex]\( r = 0.04 \)[/tex].
Here's a step-by-step guide to solve for [tex]\( P \)[/tex]:
1. Understand the problem: You're given a function [tex]\( f(t) = Pe^{rt} \)[/tex] and specific values:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
2. Substitute the known values into the function: We have
[tex]\[
246.4 = P \cdot e^{0.04 \times 4}
\][/tex]
3. Calculate [tex]\( e^{0.04 \times 4} \)[/tex]:
- Compute the exponent: [tex]\( 0.04 \times 4 = 0.16 \)[/tex]
- Now, find [tex]\( e^{0.16} \)[/tex]. The value turns out to be approximately 1.1735.
4. Solve for [tex]\( P \)[/tex]: Using the equation
[tex]\[
246.4 = P \times 1.1735
\][/tex]
- Isolate [tex]\( P \)[/tex] by dividing both sides by 1.1735:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
5. Find the value of [tex]\( P \)[/tex]: Perform the division to get [tex]\( P \approx 209.97 \)[/tex].
6. Choose the closest option: The closest option to this calculated value is:
- C. 210
Therefore, the approximate value of [tex]\( P \)[/tex] is 210.
Here's a step-by-step guide to solve for [tex]\( P \)[/tex]:
1. Understand the problem: You're given a function [tex]\( f(t) = Pe^{rt} \)[/tex] and specific values:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
2. Substitute the known values into the function: We have
[tex]\[
246.4 = P \cdot e^{0.04 \times 4}
\][/tex]
3. Calculate [tex]\( e^{0.04 \times 4} \)[/tex]:
- Compute the exponent: [tex]\( 0.04 \times 4 = 0.16 \)[/tex]
- Now, find [tex]\( e^{0.16} \)[/tex]. The value turns out to be approximately 1.1735.
4. Solve for [tex]\( P \)[/tex]: Using the equation
[tex]\[
246.4 = P \times 1.1735
\][/tex]
- Isolate [tex]\( P \)[/tex] by dividing both sides by 1.1735:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
5. Find the value of [tex]\( P \)[/tex]: Perform the division to get [tex]\( P \approx 209.97 \)[/tex].
6. Choose the closest option: The closest option to this calculated value is:
- C. 210
Therefore, the approximate value of [tex]\( P \)[/tex] is 210.
