An engineering company expects to expand its plant facilities in $10$ years at an estimated cost of $$50,000.$ To provide for the expansion, a sinking fund has been established into which equal payments are made at the beginning of every $3$ months. Interest is $9\%$ compounded quarterly.

$($a$)$ What is the size of the quarterly payment?

$($b$)$ How much of the maturity value will be payments?

$($c$)$ How much interest will the fund contain?