\( 1 \leftarrow \) Use PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) to determine the regular payment amount, rounded to the nearest dollar. The price of a small cabin is \( \$ 45,000 \). The bank requires a \( 5 \% \) down payment. The buyer is offered two mortgage options: 20-year fixed at \( 10 \% \) or 30 -year fixed at \( 10 \% \). Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 20-year option? Find the monthly payment for the 20-year option. \( \$ 413 \) (Round to the nearest dollar as needed.) Find the monthly payment for the 30-year option. \( \$ \square \) (Round to the nearest dollar as needed.)

To find the monthly payment for the 30-year mortgage option, we start by determining the amount financed after the down payment. The down payment is 5% of \$45,000, totaling \$2,250. Thus, the amount financed is \$45,000 - \$2,250 = \$42,750. Using the PMT formula with \( r = 0.10 \), \( n = 12 \), and \( t = 30 \), we can substitute these values into the formula: \[ PMT = \frac{42750 \left(\frac{0.10}{12}\right)}{\left[1-\left(1+\frac{0.10}{12}\right)^{-12 \times 30}\right]} \] Calculating this gives a monthly payment of approximately \$369. Now, let's determine the total interest for both mortgage options. For the 20-year mortgage, the total payments amount to \( 413 \times 240 = 99,120 \). The interest paid is \( 99,120 - 42,750 = 56,370 \). For the 30-year mortgage, total payments amount to \( 369 \times 360 = 132,840 \), leading to \( 132,840 - 42,750 = 90,090 \) in interest. The difference between the interest of the two options is \( 90,090 - 56,370 = 33,720 \). So, with the 30-year option, the buyer pays \$369 monthly and saves a whopping \$33,720 in interest opting for the 20-year plan!