Problem $3$: The time value of money

A housing cooperative is debating what would be the best replacement for their broken HVAC system and

have narrowed their options down to using duct tape to patch the HVAC back together or getting a new

system. The first one will cost $D(t)=D_0{t}^2$ dollars per year in repairs and future maintenance. The second

will cost $10000$$ upfront but have repair costs of $C_{0}t$ dollars per year

a$)$ Compute the present value of each option as a function of the interest rate $r$

b$)$ For which values of $D_0,C_0$ and $r$ is the first option the best one?

c$)$ After hearing about the duct taped HVAC online, the coop's insurer wants to adjust the cost of fire

insurance. If the building is worth $2$ million dollars and the risk of fire every year is $10\%,$ what is the

present value $($in terms of the interest rate$)$ of the payout in case of fire? $($Note that if an event has a

probability $p,$ the probability adjusted cost is $p($costs$))$

d$)$ If the interest rates are $8\%,$ by how much would the insurer raise the monthly premiums to avoid

taking extra risks?