A hotdog vendor must pay a monthly fee to operate a food cart in the city park. The cost in dollars of selling $x$ hundred hotdogs in a month can be approximated by $C(x)=180+65x.$ Complete parts $($a$)$ through $($d$)$ below.

$($a$)$ Find the average$-$cost function.

The average$-$cost function is $\frac{?}{\text{b}\text{a}\text{r}}(C)(x)=$

$($b$)$ Find and interpret the average cost at sales levels of $500$ hotdogs and $5500$ hotdogs.

The average cost at sales levels of $500$ hotdogs is $$\frac{,}{100}$ hotdogs.

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

The average cost at sales levels of $5500$ hotdogs is $$100$ hotdogs.

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

Interpret these results, choose the correct answer below. Take the sales level of $500$ hotdogs as an example. Select the correct choice below and fill in the answer box to complete your choice.

A$.$ At a sales level of $500$ hotdogs, the average cost is $ per $100$ hotdogs.

B$.$ At a sales level of $500$ hotdogs, if an additional $100$ hotdogs are sold, the average cost increases by approximately $ $$ per $100$ hotdogs.

$($c$)$ Find the marginal average$-$cost function.

The marginal average$-$cost function is $\frac{?}{\text{b}\text{a}\text{r}}(C{)}^{\text{'}}(x)=$

$($d$)$ Find and interpret the marginal average cost at sales levels of $500$ hotdogs and $5500$ hotdogs.

The marginal average cost at sales levels of $500$ hotdogs is $ $(100$ hotdogs