Answer :
To solve this problem, we're examining the function [tex]\( f(\ell) = P \cdot E^{r\ell} \)[/tex] and given the values [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex]. We need to find the value of [tex]\( P \)[/tex].
Here's how you can approach the problem step-by-step:
1. Identify the Given Information:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- We need to find [tex]\( P \)[/tex].
2. Understand the Function:
The function is given as [tex]\( f(\ell) = P \cdot E^{r\ell} \)[/tex].
3. Substitute the Given Values into the Function:
Since we know [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and we are to find [tex]\( P \)[/tex], plug these into the equation:
[tex]\[
191.5 = P \cdot E^{0.03 \times 3}
\][/tex]
Simplifies to:
[tex]\[
191.5 = P \cdot E^{0.09}
\][/tex]
4. Calculate [tex]\( E^{0.09} \)[/tex]:
This step involves calculating the value of the exponential term [tex]\( E^{0.09} \)[/tex]. Using the numerical approximation provided:
[tex]\[
E^{0.09} \approx 1.09417
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.09417}
\][/tex]
6. Calculate [tex]\( P \)[/tex]:
Perform the division to find the value of [tex]\( P \)[/tex]:
[tex]\[
P \approx 175.02
\][/tex]
7. Choose the Closest Answer:
Looking at the options given:
- A. 78
- B. 175
- C. 210
- D. 471
The approximate value of [tex]\( P \)[/tex] is 175 which matches option B.
Therefore, the approximate value of [tex]\( P \)[/tex] is 175, corresponding to option B.
Here's how you can approach the problem step-by-step:
1. Identify the Given Information:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- We need to find [tex]\( P \)[/tex].
2. Understand the Function:
The function is given as [tex]\( f(\ell) = P \cdot E^{r\ell} \)[/tex].
3. Substitute the Given Values into the Function:
Since we know [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and we are to find [tex]\( P \)[/tex], plug these into the equation:
[tex]\[
191.5 = P \cdot E^{0.03 \times 3}
\][/tex]
Simplifies to:
[tex]\[
191.5 = P \cdot E^{0.09}
\][/tex]
4. Calculate [tex]\( E^{0.09} \)[/tex]:
This step involves calculating the value of the exponential term [tex]\( E^{0.09} \)[/tex]. Using the numerical approximation provided:
[tex]\[
E^{0.09} \approx 1.09417
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.09417}
\][/tex]
6. Calculate [tex]\( P \)[/tex]:
Perform the division to find the value of [tex]\( P \)[/tex]:
[tex]\[
P \approx 175.02
\][/tex]
7. Choose the Closest Answer:
Looking at the options given:
- A. 78
- B. 175
- C. 210
- D. 471
The approximate value of [tex]\( P \)[/tex] is 175 which matches option B.
Therefore, the approximate value of [tex]\( P \)[/tex] is 175, corresponding to option B.
