Question $4$

Suppose a young couple takes up a housing loan of $$300,000.$ The annual interest rate on this loan is $4\%.$ The couple pays $$k$ every month for the loan.

a$)$ Write down the $1^{st}$ order ODE describing dS$/$dt where $S$ refers to the sum of outstanding loan $($ $ $),$ and $t$ refers to time in months.

b$)$ Solve the ODE to derive the relationship between $S$ and $t.$

$(2$ marks $)$

$(7$ marks $)$

c$)$ If the loan amount is fully paid in $15$ years, calculate the monthly payment $$k$ to be made by the couple. Express $k$ in dollars, rounded to the nearest cent.

$(4$ marks $)$

d$)$ Sketch of graph of $S($$$)$ vs $t($mths$),$ indicating all known values.

$(2$ marks $)$