7. As a gift to your newborn nephew, you decide to put $500 into an investment. He wili be able to access the accumulated sum when he turns 18 or at a later date if he prefers. You are comparing two investment options. One pays 5% interest compounded quarterly and the other pays 4.5% interest compounded continuously. Which option should you choose? Support your answer.
7. As a gift to your newborn nephew, you decide to put $500 into an investment. He wili be able to access the accumulated sum when he turns 18 or at a later date if he prefers. You are comparing two investment options. One pays 5% interest compounded quarterly and the other pays 4.5% interest compounded continuously. Which option should you choose? Support your answer.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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can you help me with number 7
![**Explorations**
5. **Write a model for a banking scenario where $250 is invested in an account that compounds interest continuously at a rate of 2.5%.**
- **(a) How much has accrued after 3 years?**
The formula for continuous compounding is given by:
\[
A = Pe^{rt}
\]
where \( P = 250 \), \( r = 0.025 \), and \( t = 3 \).
Calculation:
\[
250 \times e^{0.075} \approx 264.47
\]
- **(b) How long will it take to double the money?**
To find the time \( t \) when the amount doubles:
\[
2 \times 250 = 250 \times e^{0.025t}
\]
Simplifying gives:
\[
2 = e^{0.025t}
\]
Taking the natural logarithm of both sides:
\[
\ln 2 = 0.025t
\]
Solving for \( t \) gives:
\[
t = \frac{\ln 2}{0.025} \approx 27.7
\]
6. **For a banking scenario modeled by \( A = 1,000e^{0.041t} \), interpret the meaning of the constants in context. (You should be using the word "continuous" in your description.)**
- The formula represents the continuous compounding scenario where the principal amount is $1,000 and the interest rate is 4.1%. The amount \( A \) increases continuously over time \( t \).
7. **As a gift to your newborn nephew, you decide to put $500 into an investment. He will be able to access the accumulated sum when he turns 18 or at a later date if he prefers. You are comparing two investment options. One pays 5% interest compounded quarterly and the other pays 4.5% interest compounded continuously. Which option should you choose? Support your answer.**
- **Option 1: 5% interest compounded quarterly.**
Using the compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bf99a3d-f20b-4e6a-91bc-3d2554af778b%2Ffbac157c-664c-4fd3-a557-917701677e7e%2F9znt9af_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Explorations**
5. **Write a model for a banking scenario where $250 is invested in an account that compounds interest continuously at a rate of 2.5%.**
- **(a) How much has accrued after 3 years?**
The formula for continuous compounding is given by:
\[
A = Pe^{rt}
\]
where \( P = 250 \), \( r = 0.025 \), and \( t = 3 \).
Calculation:
\[
250 \times e^{0.075} \approx 264.47
\]
- **(b) How long will it take to double the money?**
To find the time \( t \) when the amount doubles:
\[
2 \times 250 = 250 \times e^{0.025t}
\]
Simplifying gives:
\[
2 = e^{0.025t}
\]
Taking the natural logarithm of both sides:
\[
\ln 2 = 0.025t
\]
Solving for \( t \) gives:
\[
t = \frac{\ln 2}{0.025} \approx 27.7
\]
6. **For a banking scenario modeled by \( A = 1,000e^{0.041t} \), interpret the meaning of the constants in context. (You should be using the word "continuous" in your description.)**
- The formula represents the continuous compounding scenario where the principal amount is $1,000 and the interest rate is 4.1%. The amount \( A \) increases continuously over time \( t \).
7. **As a gift to your newborn nephew, you decide to put $500 into an investment. He will be able to access the accumulated sum when he turns 18 or at a later date if he prefers. You are comparing two investment options. One pays 5% interest compounded quarterly and the other pays 4.5% interest compounded continuously. Which option should you choose? Support your answer.**
- **Option 1: 5% interest compounded quarterly.**
Using the compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where \(
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