The estimated total factory overhead cost and total machine hours for Department $40$ for the current year are $$250,000$ and $56,250,$ respectively. During January, the first month of the current year, actual machine hours used totaled $5,100$ and factory overhead cost incurred totaled $$22,000.$

a$.$ Determine the factory overhead rate based on machine hours. Round your answer to two decimal places.

$ per machine hour

b$.$ Journalize the entry to apply factory overhead to production in Department $40$ for January. Use the rate calculated in part $($a$).$ If an amount box does not require an entry, leave it blank. Round your answer to the nearest dollar.

c$.$ What is the balance of Factory Overhead$-$Department $40$ at January $31?$

$

d$.$ Does the balance of Factory Overhead$-$Department $40$ at January $31$ represent over$-$ or underapplied factory overhead?