Part A Which account will have a greater value after 10 years? What will the value be? $$ \begin{array}{l}\text { Account } \mathrm{A}\end{array} $$ $$ \begin{array}{l}\text { Account B } \ \text { principal: } \$ 16,000\end{array} $$ $$ \begin{array}{l}\text { annual interest: } 3 \% \ \text { compounded quarterly } \ \text { number of years: } 10\end{array} $$ $$ \begin{array}{l}\text { principal: } \$ 16,000 \ \text { annual interest: } 3 \% \ \text { compounded monthly } \ \text { number of years: } 10\end{array} $$ Account $$ \begin{array}{l}\text { Choose... }\end{array} $$ Part B What type of growth do both accounts model? A. linear most after 10 years. It will have a value of $$ \$ $$ Choose... B. exponential C neither

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UpStudy AI · Accepted Answer

Sure, let's break down both parts of your question.

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## **Part A: Comparing Account Values After 10 Years**

You have two accounts with the following details:

- **Principal (P):** \$16,000
- **Annual Interest Rate (r):** 3% (or 0.03 in decimal)
- **Time (t):** 10 years

The difference between the two accounts lies in the **compounding frequency**:

1. **Account A:** Compounded **Quarterly** (n = 4 times a year)
2. **Account B:** Compounded **Monthly** (n = 12 times a year)

### **Compound Interest Formula**

The future value $$ A $$ of an investment can be calculated using the compound interest formula:

$$
A = P \left(1 + \frac{r}{n}\right)^{nt}
$$

### **Calculations**

1. **Account A (Quarterly Compounding):**
   $$
   A = 16,000 \left(1 + \frac{0.03}{4}\right)^{4 \times 10} = 16,000 \left(1 + 0.0075\right)^{40} \approx 16,000 \times 1.34785 \approx \$21,565.57
   $$

2. **Account B (Monthly Compounding):**
   $$
   A = 16,000 \left(1 + \frac{0.03}{12}\right)^{12 \times 10} = 16,000 \left(1 + 0.0025\right)^{120} \approx 16,000 \times 1.34986 \approx \$21,597.74
   $$

### **Conclusion for Part A**

- **Greater Value:** **Account B** (Monthly Compounding)
- **Value After 10 Years:** **\$21,597.74**

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## **Part B: Type of Growth Modeled by Both Accounts**

Both Account A and Account B use the **compound interest** method to calculate future values.

### **Understanding the Growth Type**

- **Linear Growth:** Adds a fixed amount each period.
- **Exponential Growth:** Increases by a fixed percentage each period.

**Compound interest** is a classic example of **exponential growth** because the amount of interest earned each period grows based on the current total, not just the initial principal.

### **Conclusion for Part B**

- **Type of Growth:** **B. Exponential**

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**Summary:**

- **Part A:** After 10 years, **Account B** (compounded monthly) will have a greater value of **\$21,597.74**.
- **Part B:** Both accounts model **exponential growth**.
**Part A:** After 10 years, **Account B** (compounded monthly) will have a greater value of **\$21,597.74**. **Part B:** Both accounts model **exponential growth**.

Part A
Which account will have a greater value after 10 years? What will the value be?

Account
Part B
What type of growth do both accounts model?
A. linear most after 10 years. It will have a value of Choose…
B. exponential
C neither

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Part A Which account will have a greater value after 10 years? What will the value be? Account Part B What type of growth do both accounts model? A. linear most after 10 years. It will have a value of Choose… B. exponential C neither