On February $1,$ the home mortgage balance was $$278,000$ for the home owned by Amy Moore. The interest rate for the loan is $8\mathrm{.}2$ percent. Assuming that Amy makes the February monthly mortgage payment of $$2780,$ calculate the following:

$($a$)$ The amount of interest included in the February payment $($round your answer to the nearest cent$).$

$($b$)$ The amount of the monthly mortgage payment that will be used to reduce the principal balance.

$($c$)$ The new balance after Amy makes this monthly mortgage payment.

$($a$)$ Interest amount:

$($b$)$ Principal reduction:

$($c$)$ New balance: