Question:

You are in charge of marketing a new pair of sunglasses: The Ray Bob. For a sales cycle, it costs a base amount of $50,000$$$,$ and then $24$$ for each pair of glasses produced. When you set the price at $200$$ for a pair, you sell $8,000$ pairs. If you increase the price by $4$$$,$ you sell $18$ fewer pairs. Let $x$ be the number of pairs of glasses produced.

$1.$ Model the function $C(x),$ the production cost.

$2.$ Model the function $p(x),$ the price per pair of glasses.

$3.$ Model the function $R(x),$ the revenue from production.

$4.$ Model the function $P(x),$ the profit from production.

$5.$ Find the value of $x$ that maximizes the profit.

$6.$ Calculate $P(x),$ the maximum profit. Submit the answer below.

Note: Your answer must be written in decimal form with EXACTLY $2$ decimal places of precision.