Ali received a $$ \$ 2100 $$ bonus. He decided to invest it in a 5 -year certificate of deposit (CD) with an annual interest rate of $$ 1.44 \% $$ compounded quarterly Answer the questions below. Do not round any intermediate computations, and round your final answers to the nearest cent. If necessary, refer to the list of financial formulas. (a) Assuming no withdrawals are made, how much money is in Ali's account after 5 years? (b) How much interest is earned on Ali's investment after 5 years? $$ \$ \square $$

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UpStudy AI · Accepted Answer

To solve this problem, we will use the formula for compound interest, which is given by:

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

where:
- $$ A $$ is the amount of money accumulated after n years, including interest.
- $$ P $$ is the principal amount (the initial amount of money).
- $$ r $$ is the annual interest rate (decimal).
- $$ n $$ is the number of times that interest is compounded per year.
- $$ t $$ is the number of years the money is invested or borrowed.

### Given:
- $$ P = 2100 $$
- $$ r = 1.44\% = 0.0144 $$
- $$ n = 4 $$ (since the interest is compounded quarterly)
- $$ t = 5 $$

### (a) Calculate the amount in Ali's account after 5 years.

Substituting the values into the formula:

$$
A = 2100 \left(1 + \frac{0.0144}{4}\right)^{4 \times 5}
$$

Now, let's calculate this step by step.

1. Calculate $$ \frac{r}{n} $$:
   $$
   \frac{0.0144}{4} = 0.0036
   $$

2. Calculate $$ nt $$:
   $$
   4 \times 5 = 20
   $$

3. Calculate $$ A $$:
   $$
   A = 2100 \left(1 + 0.0036\right)^{20}
   $$
   $$
   A = 2100 \left(1.0036\right)^{20}
   $$

Now, we will calculate $$ (1.0036)^{20} $$ and then multiply by 2100.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$1.0036^{20}$$
- step1: Convert the expressions:
  $$\left(\frac{2509}{2500}\right)^{20}$$
- step2: Use the properties of exponents:
  $$\frac{2509^{20}}{2500^{20}}$$
Calculate or simplify the expression $$ 2100 * 1.074516 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2100\times 1.074516$$
- step1: Multiply the numbers:
  $$2256.4836$$
Now that we have calculated the necessary components, we can summarize the results:

1. $$ (1.0036)^{20} \approx 1.074516 $$
2. Therefore, the amount $$ A $$ in Ali's account after 5 years is:
   $$
   A \approx 2100 \times 1.074516 \approx 2256.48
   $$

### Final Answer for (a):
The amount in Ali's account after 5 years is approximately $$ \$ 2256.48 $$.

### (b) Calculate the interest earned on Ali's investment after 5 years.

The interest earned can be calculated by subtracting the principal from the total amount:

$$
\text{Interest} = A - P
$$
$$
\text{Interest} = 2256.48 - 2100
$$

Now, let's calculate the interest.
Calculate the value by following steps:
- step0: Calculate:
  $$2256.48-2100$$
- step1: Subtract the numbers:
  $$156.48$$
The interest earned on Ali's investment after 5 years is calculated as follows:

$$
\text{Interest} = 2256.48 - 2100 = 156.48
$$

### Final Answer for (b):
The interest earned on Ali's investment after 5 years is approximately $$ \$ 156.48 $$.

### Summary of Answers:
(a) Amount in Ali's account after 5 years: $$ \$ 2256.48 $$  
(b) Interest earned after 5 years: $$ \$ 156.48 $$
After 5 years, Ali has \$2256.48 in his account and earns \$156.48 in interest.

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