Miguel received a $$ \$ 1900 $$ bonus. He decided to invest it in a 3 -year certificate of deposit (CD) with an annual interest rate of $$ 1.32 \% $$ 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 Miguel's (b) How much interest is earned on Miguel's investment after 3 years? $$ \$ \square $$

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To solve this problem, we will use the formula for compound interest:

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

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

Given:
- $$ P = \$1900 $$ (initial investment)
- $$ r = 1.32\% = 0.0132 $$ (annual interest rate in decimal form)
- $$ n = 4 $$ (compounded quarterly)
- $$ t = 3 $$ years

(a) To find the amount of money in Miguel's account after 3 years, we substitute the given values into the compound interest formula:

$$ A = 1900 \left( 1 + \frac{0.0132}{4} \right)^{4 \cdot 3} $$

Now, we can calculate the amount of money in Miguel's account after 3 years.
Calculate the value by following steps:
- step0: Calculate:
  $$1900\left(1+\frac{0.0132}{4}\right)^{4\times 3}$$
- step1: Divide the terms:
  $$1900\left(1+\frac{33}{10000}\right)^{4\times 3}$$
- step2: Add the numbers:
  $$1900\left(\frac{10033}{10000}\right)^{4\times 3}$$
- step3: Multiply the numbers:
  $$1900\left(\frac{10033}{10000}\right)^{12}$$
- step4: Simplify:
  $$1900\times \frac{10033^{12}}{10000^{12}}$$
- step5: Rewrite the expression:
  $$100\times 19\times \frac{10033^{12}}{10000^{12}}$$
- step6: Rewrite the expression:
  $$100\times 19\times \frac{10033^{12}}{100^{24}}$$
- step7: Reduce the numbers:
  $$19\times \frac{10033^{12}}{100^{23}}$$
- step8: Multiply:
  $$\frac{19\times 10033^{12}}{100^{23}}$$
The amount of money in Miguel's account after 3 years is approximately $1976.62074.

(b) To find the interest earned on Miguel's investment after 3 years, we subtract the initial investment from the final amount:

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

Substitute the values:

$$ \text{Interest} = 1976.62074 - 1900 $$

Now, we can calculate the interest earned on Miguel's investment after 3 years.
Calculate the value by following steps:
- step0: Calculate:
  $$1976.62074-1900$$
- step1: Subtract the numbers:
  $$76.62074$$
The interest earned on Miguel's investment after 3 years is $76.62074.

Therefore, the answers are:
(a) The amount of money in Miguel's account after 3 years is approximately $1976.62074.
(b) The interest earned on Miguel's investment after 3 years is $76.62074.
(a) The amount in Miguel's account after 3 years is about $1976.62. (b) The interest earned is about $76.62.

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