Answer :
Sure! Let's solve the problem step-by-step to find the approximate value of [tex]\( P \)[/tex].
We are given a function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], where:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
We need to find the initial value [tex]\( P \)[/tex].
1. Substitute the known values into the equation:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
2. Calculate the exponent:
The exponent term [tex]\( 0.04 \cdot 4 = 0.16 \)[/tex], so we need to determine [tex]\( e^{0.16} \)[/tex].
3. Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
4. Calculate the approximate value of [tex]\( e^{0.16} \)[/tex]:
Using a calculator, find [tex]\( e^{0.16} \approx 1.17351 \)[/tex].
5. Divide to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.17351} \approx 210
\][/tex]
6. Select the closest multiple-choice answer:
The approximate value of [tex]\( P \)[/tex] is around 210.
Therefore, the correct choice is:
- B. 210
That's the step-by-step explanation for finding the approximate value of [tex]\( P \)[/tex].
We are given a function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], where:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
We need to find the initial value [tex]\( P \)[/tex].
1. Substitute the known values into the equation:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
2. Calculate the exponent:
The exponent term [tex]\( 0.04 \cdot 4 = 0.16 \)[/tex], so we need to determine [tex]\( e^{0.16} \)[/tex].
3. Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
4. Calculate the approximate value of [tex]\( e^{0.16} \)[/tex]:
Using a calculator, find [tex]\( e^{0.16} \approx 1.17351 \)[/tex].
5. Divide to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.17351} \approx 210
\][/tex]
6. Select the closest multiple-choice answer:
The approximate value of [tex]\( P \)[/tex] is around 210.
Therefore, the correct choice is:
- B. 210
That's the step-by-step explanation for finding the approximate value of [tex]\( P \)[/tex].
