Answer :
Sure, let's solve this problem step-by-step!
We are given a function [tex]\( f(t) = P e^{r \cdot t} \)[/tex]. We know that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex]. We need to find the approximate value of [tex]\( P \)[/tex].
1. Write down the function with given values:
[tex]\[
f(t) = P e^{r \cdot t}
\][/tex]
For [tex]\( t = 3 \)[/tex], the equation becomes:
[tex]\[
f(3) = P e^{0.03 \cdot 3}
\][/tex]
2. Use the given value of [tex]\( f(3) \)[/tex]:
[tex]\[
191.5 = P e^{0.03 \cdot 3}
\][/tex]
3. Calculate the exponent:
[tex]\[
0.03 \cdot 3 = 0.09
\][/tex]
So, our equation now is:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
We need to isolate [tex]\( P \)[/tex], so we rearrange the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Approximate [tex]\( e^{0.09} \)[/tex]:
Calculating [tex]\( e^{0.09} \)[/tex], we find it's approximately equal to 1.0942.
6. Calculate [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{191.5}{1.0942} \approx 175.0178
\][/tex]
7. Select the closest option:
The value of [tex]\( P \)[/tex] is approximately 175, which matches option B.
So, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].
We are given a function [tex]\( f(t) = P e^{r \cdot t} \)[/tex]. We know that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex]. We need to find the approximate value of [tex]\( P \)[/tex].
1. Write down the function with given values:
[tex]\[
f(t) = P e^{r \cdot t}
\][/tex]
For [tex]\( t = 3 \)[/tex], the equation becomes:
[tex]\[
f(3) = P e^{0.03 \cdot 3}
\][/tex]
2. Use the given value of [tex]\( f(3) \)[/tex]:
[tex]\[
191.5 = P e^{0.03 \cdot 3}
\][/tex]
3. Calculate the exponent:
[tex]\[
0.03 \cdot 3 = 0.09
\][/tex]
So, our equation now is:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
We need to isolate [tex]\( P \)[/tex], so we rearrange the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Approximate [tex]\( e^{0.09} \)[/tex]:
Calculating [tex]\( e^{0.09} \)[/tex], we find it's approximately equal to 1.0942.
6. Calculate [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{191.5}{1.0942} \approx 175.0178
\][/tex]
7. Select the closest option:
The value of [tex]\( P \)[/tex] is approximately 175, which matches option B.
So, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].
