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? \( \$ 4824.83 \) (Round to the nearest cent.) What is Cooley's payment for the loan at \( 7 \% \) interest for the next 7 years with semiannual payments? \( \$ \square \) (Round to the nearest cent.)

To calculate the different payment plans for Cooley Landscaping's loan, we can use the formula for the present value of an annuity. The formula is given by: \[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] Where: - \( PV \) is the present value (the amount borrowed, which is \( 26,000 \)). - \( P \) is the payment amount per period. - \( r \) is the interest rate per period. - \( n \) is the total number of payments. ### Step 1: Annual Payments For annual payments: - Interest rate \( r = \frac{0.07}{1} = 0.07 \) - Number of payments \( n = 7 \) We can rearrange the formula to solve for \( P \): \[ P = PV \times \left( \frac{r}{1 - (1 + r)^{-n}} \right) \] Substituting the values: \[ P = 26000 \times \left( \frac{0.07}{1 - (1 + 0.07)^{-7}} \right) \] ### Step 2: Semiannual Payments For semiannual payments: - Interest rate \( r = \frac{0.07}{2} = 0.035 \) - Number of payments \( n = 7 \times 2 = 14 \) Using the same formula for \( P \): \[ P = 26000 \times \left( \frac{0.035}{1 - (1 + 0.035)^{-14}} \right) \] ### Step 3: Quarterly Payments For quarterly payments: - Interest rate \( r = \frac{0.07}{4} = 0.0175 \) - Number of payments \( n = 7 \times 4 = 28 \) Using the same formula for \( P \): \[ P = 26000 \times \left( \frac{0.0175}{1 - (1 + 0.0175)^{-28}} \right) \] ### Step 4: Monthly Payments For monthly payments: - Interest rate \( r = \frac{0.07}{12} \approx 0.0058333 \) - Number of payments \( n = 7 \times 12 = 84 \) Using the same formula for \( P \): \[ P = 26000 \times \left( \frac{0.0058333}{1 - (1 + 0.0058333)^{-84}} \right) \] Now, let's calculate the payments for semiannual, quarterly, and monthly payments. Solve the equation by following steps: - step0: Solve for \(P\): \(26000=\frac{P\left(1-\left(1+0.0058333\right)^{-84}\right)}{0.0058333}\) - step1: Simplify: \(26000=\frac{\left(10000000\times 10058333^{84}-10000000^{85}\right)P}{58333\times 10058333^{84}}\) - step2: Swap the sides: \(\frac{\left(10000000\times 10058333^{84}-10000000^{85}\right)P}{58333\times 10058333^{84}}=26000\) - step3: Cross multiply: \(\left(10000000\times 10058333^{84}-10000000^{85}\right)P=58333\times 10058333^{84}\times 26000\) - step4: Simplify the equation: \(\left(10000000\times 10058333^{84}-10000000^{85}\right)P=1516658000\times 10058333^{84}\) - step5: Divide both sides: \(\frac{\left(10000000\times 10058333^{84}-10000000^{85}\right)P}{10000000\times 10058333^{84}-10000000^{85}}=\frac{1516658000\times 10058333^{84}}{10000000\times 10058333^{84}-10000000^{85}}\) - step6: Divide the numbers: \(P=\frac{1516658000\times 10058333^{84}}{10000000\times 10058333^{84}-10000000^{85}}\) Solve the equation \( 26000 = P * (1 - (1 + 0.0175)^{-28}) / 0.0175 \). Solve the equation by following steps: - step0: Solve for \(P\): \(26000=\frac{P\left(1-\left(1+0.0175\right)^{-28}\right)}{0.0175}\) - step1: Simplify: \(26000=\frac{\left(400\times 407^{28}-400^{29}\right)P}{7\times 407^{28}}\) - step2: Swap the sides: \(\frac{\left(400\times 407^{28}-400^{29}\right)P}{7\times 407^{28}}=26000\) - step3: Cross multiply: \(\left(400\times 407^{28}-400^{29}\right)P=7\times 407^{28}\times 26000\) - step4: Simplify the equation: \(\left(400\times 407^{28}-400^{29}\right)P=182000\times 407^{28}\) - step5: Divide both sides: \(\frac{\left(400\times 407^{28}-400^{29}\right)P}{400\times 407^{28}-400^{29}}=\frac{182000\times 407^{28}}{400\times 407^{28}-400^{29}}\) - step6: Divide the numbers: \(P=\frac{182000\times 407^{28}}{400\times 407^{28}-400^{29}}\) Solve the equation \( 26000 = P * (1 - (1 + 0.035)^{-14}) / 0.035 \). Solve the equation by following steps: - step0: Solve for \(P\): \(26000=\frac{P\left(1-\left(1+0.035\right)^{-14}\right)}{0.035}\) - step1: Simplify: \(26000=\frac{\left(200\times 207^{14}-200^{15}\right)P}{7\times 207^{14}}\) - step2: Swap the sides: \(\frac{\left(200\times 207^{14}-200^{15}\right)P}{7\times 207^{14}}=26000\) - step3: Cross multiply: \(\left(200\times 207^{14}-200^{15}\right)P=7\times 207^{14}\times 26000\) - step4: Simplify the equation: \(\left(200\times 207^{14}-200^{15}\right)P=182000\times 207^{14}\) - step5: Divide both sides: \(\frac{\left(200\times 207^{14}-200^{15}\right)P}{200\times 207^{14}-200^{15}}=\frac{182000\times 207^{14}}{200\times 207^{14}-200^{15}}\) - step6: Divide the numbers: \(P=\frac{182000\times 207^{14}}{200\times 207^{14}-200^{15}}\) Calculate or simplify the expression \( 26000 * (0.035 / (1 - (1 + 0.035)^{-14})) \). Calculate the value by following steps: - step0: Calculate: \(26000\left(\frac{0.035}{\left(1-\left(1+0.035\right)^{-14}\right)}\right)\) - step1: Remove the parentheses: \(26000\left(\frac{0.035}{1-\left(1+0.035\right)^{-14}}\right)\) - step2: Add the numbers: \(26000\left(\frac{0.035}{1-1.035^{-14}}\right)\) - step3: Convert the expressions: \(26000\left(\frac{0.035}{1-\left(\frac{207}{200}\right)^{-14}}\right)\) - step4: Subtract the numbers: \(26000\left(\frac{0.035}{\frac{207^{14}-200^{14}}{207^{14}}}\right)\) - step5: Divide the terms: \(26000\times \frac{7\times 207^{14}}{200\times 207^{14}-200^{15}}\) - step6: Multiply: \(\frac{26000\times 7\times 207^{14}}{200\times 207^{14}-200^{15}}\) - step7: Multiply: \(\frac{182000\times 207^{14}}{200\times 207^{14}-200^{15}}\) Calculate or simplify the expression \( 26000 * (0.0175 / (1 - (1 + 0.0175)^{-28})) \). Calculate the value by following steps: - step0: Calculate: \(26000\left(\frac{0.0175}{\left(1-\left(1+0.0175\right)^{-28}\right)}\right)\) - step1: Remove the parentheses: \(26000\left(\frac{0.0175}{1-\left(1+0.0175\right)^{-28}}\right)\) - step2: Add the numbers: \(26000\left(\frac{0.0175}{1-1.0175^{-28}}\right)\) - step3: Convert the expressions: \(26000\left(\frac{0.0175}{1-\left(\frac{407}{400}\right)^{-28}}\right)\) - step4: Subtract the numbers: \(26000\left(\frac{0.0175}{\frac{407^{28}-400^{28}}{407^{28}}}\right)\) - step5: Divide the terms: \(26000\times \frac{7\times 407^{28}}{400\times 407^{28}-400^{29}}\) - step6: Multiply: \(\frac{26000\times 7\times 407^{28}}{400\times 407^{28}-400^{29}}\) - step7: Multiply: \(\frac{182000\times 407^{28}}{400\times 407^{28}-400^{29}}\) Calculate or simplify the expression \( 26000 * (0.0058333 / (1 - (1 + 0.0058333)^{-84})) \). Calculate the value by following steps: - step0: Calculate: \(26000\left(\frac{0.0058333}{\left(1-\left(1+0.0058333\right)^{-84}\right)}\right)\) - step1: Remove the parentheses: \(26000\left(\frac{0.0058333}{1-\left(1+0.0058333\right)^{-84}}\right)\) - step2: Add the numbers: \(26000\left(\frac{0.0058333}{1-1.0058333^{-84}}\right)\) - step3: Convert the expressions: \(26000\left(\frac{0.0058333}{1-\left(\frac{10058333}{10000000}\right)^{-84}}\right)\) - step4: Subtract the numbers: \(26000\left(\frac{0.0058333}{\frac{10058333^{84}-10000000^{84}}{10058333^{84}}}\right)\) - step5: Divide the terms: \(26000\times \frac{58333\times 10058333^{84}}{10000000\times 10058333^{84}-10000000^{85}}\) - step6: Multiply: \(\frac{26000\times 58333\times 10058333^{84}}{10000000\times 10058333^{84}-10000000^{85}}\) - step7: Multiply: \(\frac{1516658000\times 10058333^{84}}{10000000\times 10058333^{84}-10000000^{85}}\) Now that we have calculated the payments for the different payment plans, here are the results: 1. **Semiannual Payments**: - Payment: \( P \approx 2380.84 \) 2. **Quarterly Payments**: - Payment: \( P \approx 1182.52 \) 3. **Monthly Payments**: - Payment: \( P \approx 392.41 \) ### Summary of Payments: - **Annual Payments**: \( \$ 4824.83 \) - **Semiannual Payments**: \( \$ 2380.84 \) - **Quarterly Payments**: \( \$ 1182.52 \) - **Monthly Payments**: \( \$ 392.41 \) All amounts are rounded to the nearest cent.