$$ \begin{array}{l}\$ 40,500 ext { are deposited into an account } \ ext { with a } 3.8 \% ext { interest rate, compounded } \ ext { monthly ( } 12 ext { times per year). } \ ext { Find the accumulated amount after } \ 20 ext { years. } \ ext { Hint: } A=P\left(1+\frac{r}{k} ight)^{k t}\end{array} $$ Round your answer to the nearest cent (hundredth

Question

$$ \begin{array}{l}\$ 40,500 	ext { are deposited into an account } \ 	ext { with a } 3.8 \% 	ext { interest rate, compounded } \ 	ext { monthly ( } 12 	ext { times per year). } \ 	ext { Find the accumulated amount after } \ 20 	ext { years. } \ 	ext { Hint: } A=P\left(1+\frac{r}{k}ight)^{k t}\end{array} $$ Round your answer to the nearest cent (hundredth

UpStudy AI · Accepted Answer

To find the accumulated amount after 20 years with monthly compounding interest, we'll use the compound interest formula:

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

Where:
- $$ A $$ = the amount of money accumulated after $$ t $$ years, including interest.
- $$ P $$ = the principal amount (\$40,500).
- $$ r $$ = annual interest rate (3.8\% or 0.038).
- $$ k $$ = number of times the interest is compounded per year (12 for monthly).
- $$ t $$ = number of years (20).

### Step-by-Step Calculation:

1. **Plug in the known values:**
   $$
   A = 40,500 \times \left(1 + \frac{0.038}{12}\right)^{12 \times 20}
   $$

2. **Calculate the monthly interest rate:**
   $$
   \frac{0.038}{12} = 0.0031666667
   $$
   So,
   $$
   1 + 0.0031666667 = 1.0031666667
   $$

3. **Calculate the total number of compounding periods:**
   $$
   12 \times 20 = 240
   $$

4. **Raise the base to the power of the number of periods:**
   $$
   1.0031666667^{240} \approx 2.1340
   $$
   
5. **Multiply by the principal amount:**
   $$
   40,500 \times 2.1340 \approx 86,427.00
   $$

### Final Answer:

After 20 years, the accumulated amount is **\$86,427.00**.
After 20 years, the accumulated amount is \$86,427.00.

Round your answer to the nearest cent (hundredth

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