Elizabeth went on a fabulous vacation in May and racked up a lot of charges on

her credit card. When it came time to pay her June credit card bill, she left a

balance of $$1180\mathrm{.}00.$ Elizabeth's credit card billing cycle runs from the $19$th of

each month to the $18$ th of the next month, and her interest rate is $19\mathrm{.}5\%.$ She

started the billing cycle June $19-$July $18$ with a previous balance of $$1180\mathrm{.}00.$ In

addition she made three purchases, with the dates and amounts shown in the

table to the right. On July $15$ she made an online payment of $$510\mathrm{.}00$ that was

credited to her balance the same day. Complete parts $($a$)$ through $($c$).$

$($a$)$ Find the average daily balance on the credit card account for the billing cycle June $19-$July $18.$

The average daily balance for the billing cycle is $

$($Round to the nearest cent as needed.$)$