Answer :
To find the approximate value of [tex]\( P \)[/tex], we use the given information along with the function:
[tex]\[
f(t) = P e^{rt}
\][/tex]
We know that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex]. We can substitute these values into the equation to find [tex]\( P \)[/tex]:
1. Start with the equation:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/tex]
3. Use a calculator to find [tex]\( e^{0.25} \)[/tex]. This is approximately [tex]\( 1.284 \)[/tex].
4. Substitute this value back into the equation:
[tex]\[
288.9 = P \times 1.284
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
6. Calculating the division, we find:
[tex]\[
P \approx 225
\][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 225. Therefore, the correct answer is option C: 225.
[tex]\[
f(t) = P e^{rt}
\][/tex]
We know that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex]. We can substitute these values into the equation to find [tex]\( P \)[/tex]:
1. Start with the equation:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/tex]
3. Use a calculator to find [tex]\( e^{0.25} \)[/tex]. This is approximately [tex]\( 1.284 \)[/tex].
4. Substitute this value back into the equation:
[tex]\[
288.9 = P \times 1.284
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
6. Calculating the division, we find:
[tex]\[
P \approx 225
\][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 225. Therefore, the correct answer is option C: 225.
