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Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] given the information. Let's follow these steps:
1. Understand the given information:
- The function is [tex]\( f(t) = P \times e^{r \times t} \)[/tex].
- We're given that [tex]\( f(5) = 288.9 \)[/tex] and the rate [tex]\( r = 0.05 \)[/tex].
2. Set up the equation:
When [tex]\( t = 5 \)[/tex], the equation becomes:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]
3. Calculate the exponent:
- First, calculate the exponent part: [tex]\( 0.05 \times 5 = 0.25 \)[/tex].
4. Solve for [tex]\( P \)[/tex]:
- Using the equation [tex]\( 288.9 = P \times e^{0.25} \)[/tex], we need to isolate [tex]\( P \)[/tex].
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
5. Determine the value of [tex]\( P \)[/tex]:
- Approximate [tex]\( e^{0.25} \)[/tex] using a calculator. After calculating, you find that [tex]\( e^{0.25} \approx 1.284 \)[/tex].
- Substitute back into the equation:
[tex]\[
P = \frac{288.9}{1.284} \approx 225
\][/tex]
The approximate value of [tex]\( P \)[/tex] is closest to 225. Therefore, the correct answer is:
D. 225
1. Understand the given information:
- The function is [tex]\( f(t) = P \times e^{r \times t} \)[/tex].
- We're given that [tex]\( f(5) = 288.9 \)[/tex] and the rate [tex]\( r = 0.05 \)[/tex].
2. Set up the equation:
When [tex]\( t = 5 \)[/tex], the equation becomes:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]
3. Calculate the exponent:
- First, calculate the exponent part: [tex]\( 0.05 \times 5 = 0.25 \)[/tex].
4. Solve for [tex]\( P \)[/tex]:
- Using the equation [tex]\( 288.9 = P \times e^{0.25} \)[/tex], we need to isolate [tex]\( P \)[/tex].
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
5. Determine the value of [tex]\( P \)[/tex]:
- Approximate [tex]\( e^{0.25} \)[/tex] using a calculator. After calculating, you find that [tex]\( e^{0.25} \approx 1.284 \)[/tex].
- Substitute back into the equation:
[tex]\[
P = \frac{288.9}{1.284} \approx 225
\][/tex]
The approximate value of [tex]\( P \)[/tex] is closest to 225. Therefore, the correct answer is:
D. 225
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