4. (18 points) (a) [8 points] Derive a formula for the monthly mortgage payment x, expressed in terms of the principal amount P, monthly interest rate r, and total number of months of the loan n. Show your work and the detailed steps. The answer for this problem (a) is a formula. (b) [4 points] Given the following data: The principal amount (i.e., the total loan) is P = $400,000, the annual interest rate is 3.0% (converted into the monthly rate 0.25%), and the loan is to be paid off in 30 years (equivalent to 360 months). Use the above derived formula and the data to compute the monthly mortgage payment x by a calculator or R. The answer should be an amount of money per month. You do not need to submit the R code for this problem. (c) [3 points] If the annual rate is reduced to 2.95% in the above data, what is the monthly mort- gage payment? (d) [3 points) If the principal is increased to P = $470,000, the annual rate is 2.875%, and the loan period is still 30 years, what is the the monthly mortgage payment now? 4. (18 points) (a) [8 points] Derive a formula for the monthly mortgage payment x, expressed in terms of the principal amount P, monthly interest rate r, and total number of months of the loan n. Show your work and the detailed steps. The answer for this problem (a) is a formula. (b) [4 points] Given the following data: The principal amount (i.e., the total loan) is P = $400,000, the annual interest rate is 3.0% (converted into the monthly rate 0.25%), and the loan is to be paid off in 30 years (equivalent to 360 months). Use the above derived formula and the data to compute the monthly mortgage payment x by a calculator or R. The answer should be an amount of money per month. You do not need to submit the R code for this problem. (c) [3 points] If the annual rate is reduced to 2.95% in the above data, what is the monthly mort- gage payment? (d) [3 points) If the principal is increased to P = $470,000, the annual rate is 2.875%, and the loan period is still 30 years, what is the the monthly mortgage payment now