Smart Fuels offers discount home heating services. They deliver furnace oil at competitive prices

and service furnaces at a lower labour rate than their major competitors. They are encouraging

homeowners to replace their furnaces with a super efficient furnace. The purchase and

installation of the super$-$efficient furnace is $$4,000.$ Estimated annual savings are substantial. At

current oil prices, an older home with poor insulation can expect to save $$800$ per year. Smart

Fuels argues that the new furnace will pay for itself in $5$ years $($which is $3$ times across the $15-$

year life expectancy of the furnace$).$

$($a$)$ Suppose you are a homeowner and must borrow the funds to purchase the furnace. You are using

a line of credit that is charged $9\%$ compounded annually. How much would $$4,000$ today grow to

in $5$ years at $9\%$ compounded annually?

$($b$)$ At the end of each year, you put the $$800$ you have saved towards the furnace. How much would

these payments accumulate to in $5$ years, if they were compounded at $9\%$ annually? How much

would you therefore owe at the end of $5$ years?

$($c$)$ How much would $$4,000$ today grow to in $15$ years at $9\%$ compounded annually?

$($d$)$ How much would $$800$ per year, paid at the end of the year, grow to $9\%$ compounded annually?