Image Manufacturing, Inc., sells parts for photographic equipment. For example, one developing machine component sells for $$2$ each to a company buying $15,000$ pieces per month. Image Manufacturing gives an incentive for higher volume. It offers a $5\%$ rebate on orders of $20,000$ pieces per month, or a per$-$piece rebate for orders of at least $22,000$ pieces per month.

Suppose a company currently orders $12,000$ parts per month from Image Manufacturing at $$2$ each, which is about half of what they buy each month from other suppliers. If they move $8000$ pieces per month from another company to Image Manufacturing, what will be their rebate on the total order? What will be the discounted cost per piece?

The rebate on the order is $$2,000.$

The discounted cost per piece is $$1\mathrm{.}90.($Type an integer or a decimal.$)$

$2.$ If the customer increases its order to $22,000$ pieces per month and negotiates a $14$ cents$-$per$-$piece trade discount, what will be the rebate? What is the percent of the discount?

The rebate is $$3,080.$

The percent of the discount is $7\mathrm{.}0\%.($Round to the nearest tenth as needed.$)$

$3.$ In addition to the $14$ cent$-$per$-$piece trade discount, the customer also receives a $\frac{1}{2}\%$ cash discount. Calculate the rebate and cash discount on a $30,000-$piece$-$per$-$month order, and then find the net price. The cash discount of $\frac{1}{2}\%$ is taken after the rebate is applied.

The rebate on the order is $$.$