Answer :
To find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], we are given that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex]. We need to use this information to solve for [tex]\( P \)[/tex].
Here's how we can do it step-by-step:
1. Understand the Given Information:
- The function is [tex]\( f(t) = P e^{rt} \)[/tex].
- We have [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex].
2. Substitute the Known Values into the Function:
- Substitute [tex]\( t = 3 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( f(3) = 191.5 \)[/tex] into the function:
[tex]\[
191.5 = P \times e^{0.03 \times 3}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- First, calculate the exponent:
[tex]\[
0.03 \times 3 = 0.09
\][/tex]
- Then, calculate [tex]\( e^{0.09} \)[/tex]. This is approximately 1.0942.
- Substitute this back into the equation:
[tex]\[
191.5 = P \times 1.0942
\][/tex]
- Solve for [tex]\( P \)[/tex] by dividing both sides by 1.0942:
[tex]\[
P \approx \frac{191.5}{1.0942} \approx 175
\][/tex]
4. Check Available Options:
- Based on our calculation, the approximate value of [tex]\( P \)[/tex] is closest to 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{175}\)[/tex].
Here's how we can do it step-by-step:
1. Understand the Given Information:
- The function is [tex]\( f(t) = P e^{rt} \)[/tex].
- We have [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex].
2. Substitute the Known Values into the Function:
- Substitute [tex]\( t = 3 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( f(3) = 191.5 \)[/tex] into the function:
[tex]\[
191.5 = P \times e^{0.03 \times 3}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- First, calculate the exponent:
[tex]\[
0.03 \times 3 = 0.09
\][/tex]
- Then, calculate [tex]\( e^{0.09} \)[/tex]. This is approximately 1.0942.
- Substitute this back into the equation:
[tex]\[
191.5 = P \times 1.0942
\][/tex]
- Solve for [tex]\( P \)[/tex] by dividing both sides by 1.0942:
[tex]\[
P \approx \frac{191.5}{1.0942} \approx 175
\][/tex]
4. Check Available Options:
- Based on our calculation, the approximate value of [tex]\( P \)[/tex] is closest to 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{175}\)[/tex].
