Answer :
Sure! Let's solve the problem step-by-step to find the approximate value of [tex]\( P \)[/tex].
We are given the function:
[tex]\[ f(t) = P e^{rt} \][/tex]
And we know:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
Our goal is to find the value of [tex]\( P \)[/tex].
1. Substitute the known values into the function:
Since [tex]\( f(3) = 191.5 \)[/tex], we substitute 3 for [tex]\( t \)[/tex] and 191.5 for [tex]\( f(3) \)[/tex]:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
2. Calculate the exponent:
The exponent in the equation is [tex]\( r \times 3 = 0.03 \times 3 = 0.09 \)[/tex].
3. Evaluate the exponential expression:
Calculate [tex]\( e^{0.09} \)[/tex]. This approximately equals 1.0941742837052104.
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation [tex]\( 191.5 = P \times 1.0941742837052104 \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.0941742837052104}
\][/tex]
This gives us [tex]\( P \approx 175.01782197944019 \)[/tex].
5. Determine the approximate value:
The closest answer choice to 175.01782 is 175.
So, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].
We are given the function:
[tex]\[ f(t) = P e^{rt} \][/tex]
And we know:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
Our goal is to find the value of [tex]\( P \)[/tex].
1. Substitute the known values into the function:
Since [tex]\( f(3) = 191.5 \)[/tex], we substitute 3 for [tex]\( t \)[/tex] and 191.5 for [tex]\( f(3) \)[/tex]:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
2. Calculate the exponent:
The exponent in the equation is [tex]\( r \times 3 = 0.03 \times 3 = 0.09 \)[/tex].
3. Evaluate the exponential expression:
Calculate [tex]\( e^{0.09} \)[/tex]. This approximately equals 1.0941742837052104.
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation [tex]\( 191.5 = P \times 1.0941742837052104 \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.0941742837052104}
\][/tex]
This gives us [tex]\( P \approx 175.01782197944019 \)[/tex].
5. Determine the approximate value:
The closest answer choice to 175.01782 is 175.
So, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].
