Answer :
To solve the problem, we need to determine the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
Here's the step-by-step solution:
1. Understand the given function and values:
- The function is [tex]\( f(t) = P e^{rt} \)[/tex].
- Given values are [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
2. Substitute the given values into the function:
- We need to find [tex]\( P \)[/tex].
- Substitute [tex]\( t = 3 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex], and [tex]\( r = 0.03 \)[/tex] into the function:
[tex]\[
191.5 = P \cdot e^{0.03 \cdot 3}
\][/tex]
3. Simplify the exponent calculation:
- Calculate the exponent [tex]\( 0.03 \cdot 3 \)[/tex]:
[tex]\[
0.03 \cdot 3 = 0.09
\][/tex]
4. Rewrite the equation with the exponent value:
- Substitute [tex]\( e^{0.09} \)[/tex] into the equation:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
- To isolate [tex]\( P \)[/tex], we need to divide both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
6. Calculate the value of [tex]\( e^{0.09} \)[/tex]:
- The approximate value of [tex]\( e^{0.09} \)[/tex] is a constant that can be evaluated to find the value of [tex]\( P \)[/tex].
7. Perform the final division:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\approx 175.01782197944019
\][/tex]
The closest value to [tex]\( 175.01782197944019 \)[/tex] among the given choices is:
- A. 471
- B. 210
- C. 78
- D. 175
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].
Here's the step-by-step solution:
1. Understand the given function and values:
- The function is [tex]\( f(t) = P e^{rt} \)[/tex].
- Given values are [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
2. Substitute the given values into the function:
- We need to find [tex]\( P \)[/tex].
- Substitute [tex]\( t = 3 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex], and [tex]\( r = 0.03 \)[/tex] into the function:
[tex]\[
191.5 = P \cdot e^{0.03 \cdot 3}
\][/tex]
3. Simplify the exponent calculation:
- Calculate the exponent [tex]\( 0.03 \cdot 3 \)[/tex]:
[tex]\[
0.03 \cdot 3 = 0.09
\][/tex]
4. Rewrite the equation with the exponent value:
- Substitute [tex]\( e^{0.09} \)[/tex] into the equation:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
- To isolate [tex]\( P \)[/tex], we need to divide both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
6. Calculate the value of [tex]\( e^{0.09} \)[/tex]:
- The approximate value of [tex]\( e^{0.09} \)[/tex] is a constant that can be evaluated to find the value of [tex]\( P \)[/tex].
7. Perform the final division:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\approx 175.01782197944019
\][/tex]
The closest value to [tex]\( 175.01782197944019 \)[/tex] among the given choices is:
- A. 471
- B. 210
- C. 78
- D. 175
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].
