4. (7 pts) Suppose you want to have $$ \$ 24,000 $$ in 4 years from now to buy a car. How much money should you invest now if the account earns $$ 7.2 \% $$ compounded quarterly? Clearly state the solution to the problem.

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We start with the compound interest formula

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

where  
$$ A $$ is the amount in the future,  
$$ P $$ is the principal (the amount invested now),  
$$ r $$ is the annual interest rate (in decimal form),  
$$ n $$ is the number of compounding periods per year, and  
$$ t $$ is the number of years.

Given:  
$$ A = 24000 $$,  
$$ r = 0.072 $$,  
$$ n = 4 $$ and  
$$ t = 4 $$.

We solve for $$ P $$:

$$
P = \frac{A}{\left(1+\frac{r}{n}\right)^{nt}} = \frac{24000}{\left(1+\frac{0.072}{4}\right)^{4 \times 4}}
$$

First, compute the periodic interest rate:

$$
1+\frac{0.072}{4} = 1 + 0.018 = 1.018
$$

Next, compute the total number of compounding periods:

$$
4 \times 4 = 16
$$

So we have:

$$
P = \frac{24000}{(1.018)^{16}}
$$

Now, evaluate $$(1.018)^{16}$$. Using a calculator, one obtains approximately

$$
(1.018)^{16} \approx 1.3308
$$

Thus,

$$
P \approx \frac{24000}{1.3308} \approx 18034.45
$$

Rounded to the nearest dollar, you should invest approximately

$$
\boxed{\$18,034}
$$
You should invest approximately \$18,034 now to have \$24,000 in 4 years with a 7.2% interest rate compounded quarterly.

(7 pts) Suppose you want to have in 4 years from now to buy a car. How much money should you invest now if the account earns
compounded quarterly? Clearly state the solution to the problem.

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(7 pts) Suppose you want to have in 4 years from now to buy a car. How much money should you invest now if the account earns compounded quarterly? Clearly state the solution to the problem.