Present value with periodic rates. Cooley Landscaping needs to borrow \( \$ 26,000 \) for a new front-end dirt loader. The bank is willing to loan the money at \( 7 \% \) interest for the next 7 years with annual, semiannual, quarterly, or monthly payments. What are the different payments that Cooley Landscaping could choose for these different payment plans? What is Cooley's payment for the loan at \( 7 \% \) interest for the next 7 years with annual payments? \( \$ \square \) (Round to the nearest cent.)

To find the different payments that Cooley Landscaping could choose for the different payment plans, we need to calculate the present value of the loan using the formula for present value with periodic rates. The formula for present value with periodic rates is given by: \[ PV = \frac{PMT}{r} \left( 1 - \left(1 + \frac{r}{n}\right)^{-nt} \right) \] Where: - \( PV \) is the present value of the loan - \( PMT \) is the periodic payment - \( r \) is the annual interest rate - \( n \) is the number of times the interest is compounded per year - \( t \) is the number of years Given: - \( PV = \$26,000 \) - \( r = 7\% = 0.07 \) - \( t = 7 \) years We need to calculate the present value for different payment plans: 1. Annual payments 2. Semiannual payments 3. Quarterly payments 4. Monthly payments Let's calculate the present value for each payment plan. Calculate the value by following steps: - step0: Calculate: \(\frac{26000\left(1-\left(1+\frac{0.07}{1}\right)^{-7}\right)}{\left(\frac{0.07}{1}\right)}\) - step1: Remove the parentheses: \(\frac{26000\left(1-\left(1+\frac{0.07}{1}\right)^{-7}\right)}{\frac{0.07}{1}}\) - step2: Divide the terms: \(\frac{26000\left(1-\left(1+\frac{7}{100}\right)^{-7}\right)}{\frac{0.07}{1}}\) - step3: Add the numbers: \(\frac{26000\left(1-\left(\frac{107}{100}\right)^{-7}\right)}{\frac{0.07}{1}}\) - step4: Subtract the numbers: \(\frac{26000\times \frac{107^{7}-100^{7}}{107^{7}}}{\frac{0.07}{1}}\) - step5: Divide the terms: \(\frac{26000\times \frac{107^{7}-100^{7}}{107^{7}}}{\frac{7}{100}}\) - step6: Multiply the numbers: \(\frac{\frac{26000\times 107^{7}-26000\times 100^{7}}{107^{7}}}{\frac{7}{100}}\) - step7: Multiply by the reciprocal: \(\frac{26000\times 107^{7}-26000\times 100^{7}}{107^{7}}\times \frac{100}{7}\) - step8: Multiply the fractions: \(\frac{\left(26000\times 107^{7}-26000\times 100^{7}\right)\times 100}{107^{7}\times 7}\) - step9: Multiply: \(\frac{2600000\times 107^{7}-2600000\times 100^{7}}{107^{7}\times 7}\) The present value for annual payments is approximately $140,121.52. Now, let's calculate the present value for semiannual payments. Calculate the value by following steps: - step0: Calculate: \(\frac{26000\left(1-\left(1+\frac{0.07}{2}\right)^{-2\times 7}\right)}{\left(\frac{0.07}{2}\right)}\) - step1: Remove the parentheses: \(\frac{26000\left(1-\left(1+\frac{0.07}{2}\right)^{-2\times 7}\right)}{\frac{0.07}{2}}\) - step2: Divide the terms: \(\frac{26000\left(1-\left(1+\frac{7}{200}\right)^{-2\times 7}\right)}{\frac{0.07}{2}}\) - step3: Add the numbers: \(\frac{26000\left(1-\left(\frac{207}{200}\right)^{-2\times 7}\right)}{\frac{0.07}{2}}\) - step4: Multiply the numbers: \(\frac{26000\left(1-\left(\frac{207}{200}\right)^{-14}\right)}{\frac{0.07}{2}}\) - step5: Subtract the numbers: \(\frac{26000\times \frac{207^{14}-200^{14}}{207^{14}}}{\frac{0.07}{2}}\) - step6: Divide the terms: \(\frac{26000\times \frac{207^{14}-200^{14}}{207^{14}}}{\frac{7}{200}}\) - step7: Multiply the numbers: \(\frac{\frac{26000\times 207^{14}-26000\times 200^{14}}{207^{14}}}{\frac{7}{200}}\) - step8: Multiply by the reciprocal: \(\frac{26000\times 207^{14}-26000\times 200^{14}}{207^{14}}\times \frac{200}{7}\) - step9: Multiply the fractions: \(\frac{\left(26000\times 207^{14}-26000\times 200^{14}\right)\times 200}{207^{14}\times 7}\) - step10: Multiply: \(\frac{5200000\times 207^{14}-5200000\times 200^{14}}{207^{14}\times 7}\) The present value for semiannual payments is approximately $283,933.53. Next, let's calculate the present value for quarterly payments. Calculate the value by following steps: - step0: Calculate: \(\frac{26000\left(1-\left(1+\frac{0.07}{4}\right)^{-4\times 7}\right)}{\left(\frac{0.07}{4}\right)}\) - step1: Remove the parentheses: \(\frac{26000\left(1-\left(1+\frac{0.07}{4}\right)^{-4\times 7}\right)}{\frac{0.07}{4}}\) - step2: Divide the terms: \(\frac{26000\left(1-\left(1+\frac{7}{400}\right)^{-4\times 7}\right)}{\frac{0.07}{4}}\) - step3: Add the numbers: \(\frac{26000\left(1-\left(\frac{407}{400}\right)^{-4\times 7}\right)}{\frac{0.07}{4}}\) - step4: Multiply the numbers: \(\frac{26000\left(1-\left(\frac{407}{400}\right)^{-28}\right)}{\frac{0.07}{4}}\) - step5: Subtract the numbers: \(\frac{26000\times \frac{407^{28}-400^{28}}{407^{28}}}{\frac{0.07}{4}}\) - step6: Divide the terms: \(\frac{26000\times \frac{407^{28}-400^{28}}{407^{28}}}{\frac{7}{400}}\) - step7: Multiply the numbers: \(\frac{\frac{26000\times 407^{28}-26000\times 400^{28}}{407^{28}}}{\frac{7}{400}}\) - step8: Multiply by the reciprocal: \(\frac{26000\times 407^{28}-26000\times 400^{28}}{407^{28}}\times \frac{400}{7}\) - step9: Multiply the fractions: \(\frac{\left(26000\times 407^{28}-26000\times 400^{28}\right)\times 400}{407^{28}\times 7}\) - step10: Multiply: \(\frac{10400000\times 407^{28}-10400000\times 400^{28}}{407^{28}\times 7}\) The present value for quarterly payments is approximately $571,660.82. Finally, let's calculate the present value for monthly payments. Calculate the value by following steps: - step0: Calculate: \(\frac{26000\left(1-\left(1+\frac{0.07}{12}\right)^{-12\times 7}\right)}{\left(\frac{0.07}{12}\right)}\) - step1: Remove the parentheses: \(\frac{26000\left(1-\left(1+\frac{0.07}{12}\right)^{-12\times 7}\right)}{\frac{0.07}{12}}\) - step2: Divide the terms: \(\frac{26000\left(1-\left(1+\frac{7}{1200}\right)^{-12\times 7}\right)}{\frac{0.07}{12}}\) - step3: Add the numbers: \(\frac{26000\left(1-\left(\frac{1207}{1200}\right)^{-12\times 7}\right)}{\frac{0.07}{12}}\) - step4: Multiply the numbers: \(\frac{26000\left(1-\left(\frac{1207}{1200}\right)^{-84}\right)}{\frac{0.07}{12}}\) - step5: Subtract the numbers: \(\frac{26000\times \frac{1207^{84}-1200^{84}}{1207^{84}}}{\frac{0.07}{12}}\) - step6: Divide the terms: \(\frac{26000\times \frac{1207^{84}-1200^{84}}{1207^{84}}}{\frac{7}{1200}}\) - step7: Multiply the numbers: \(\frac{\frac{26000\times 1207^{84}-26000\times 1200^{84}}{1207^{84}}}{\frac{7}{1200}}\) - step8: Multiply by the reciprocal: \(\frac{26000\times 1207^{84}-26000\times 1200^{84}}{1207^{84}}\times \frac{1200}{7}\) - step9: Multiply the fractions: \(\frac{\left(26000\times 1207^{84}-26000\times 1200^{84}\right)\times 1200}{1207^{84}\times 7}\) - step10: Multiply: \(\frac{31200000\times 1207^{84}-31200000\times 1200^{84}}{1207^{84}\times 7}\) The present value for monthly payments is approximately $1,722,689.41. Therefore, the different payments that Cooley Landscaping could choose for the different payment plans are: - Annual payments: $140,121.52 - Semiannual payments: $283,933.53 - Quarterly payments: $571,660.82 - Monthly payments: $1,722,689.41 The payment for the loan at 7% interest for the next 7 years with annual payments is $140,121.52.