Decide which of the two plans will provide a better yield.
Plan​ A: ​$48 comma 000 invested for 3 years at​ 1.5%, compounded quarterly 
Plan​ B: ​$48 comma 000 invested for 3 years at​ 1.4%, compounded continuously
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
  
A.
Plan B is better by ​$
  
enter your response here.
​(Simplify your answer. Type an integer or a decimal. Round to the nearest cent as​ needed.)
B.
Plan A is better by ​$
  
enter your response here.
​(Simplify your answer. Type an integer or a decimal. Round to the nearest cent as​ needed.)
C.
Both plans are same.

To determine which investment plan provides a better yield, we'll calculate the final value of each plan using the appropriate compound interest formulas.

### Plan A: Compound Interest Quarterly
The formula for compound interest is:

\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]
Where:
- $ P = 48,000 $ (initial investment)
- $ r = 1.5\% = 0.015 $ (annual interest rate)
- $ n = 4 $ (quarterly compounding)
- $ t = 3 $ (number of years)

Substitute these values into the formula:

\[
A_A = 48,000 \left( 1 + \frac{0.015}{4} \right)^{4 \times 3}
\]

### Plan B: Compound Interest Continuously
The formula for continuous compounding is:

\[
A = P e^{rt}
\]
Where:
- $ P = 48,000 $ (initial investment)
- $ r = 1.4\% = 0.014 $ (annual interest rate)
- $ t = 3 $ (number of years)
- $ e $ is Euler's number (approximately 2.71828)

Substitute these values into the formula:

\[
A_B = 48,000 \times e^{0.014 \times 3}
\]

Now I'll calculate the values for both Plan A and Plan B.The final amounts for both plans are as follows:

- **Plan A** (1.5% compounded quarterly): \$50,205.11
- **Plan B** (1.4% compounded continuously): \$50,058.93

The difference is approximately \$146.18, so **Plan A** provides a better yield by \$146.18.

Would you like further details or have any questions?

### 5 related questions:
1. How does the frequency of compounding affect interest calculations?
2. What is the formula for determining compound interest when compounding annually?
3. How does continuous compounding differ from regular compound interest?
4. How can you compare different investment plans with varying interest rates and periods?
5. What is Euler's number, and why is it important in continuous compounding?

### Tip:
Compounding more frequently results in more interest accumulation over time, even with the same annual interest rate.

Decide which of the two plans will provide a better yield. Plan A: $48,000 invested for 3 years at 1.5%, compounded quarterly. Plan B: $48,000 invested for 3 years at 1.4%, compounded continuously. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Learn how to compare two investment plans using compound interest and continuous compounding formulas. Plan A offers $48,000 at 1.5% compounded quarterly, while Plan B offers $48,000 at 1.4% compounded continuously over 3 years. Determine which plan yields better returns and by how much. Suitable for Grades 10-12.

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