The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year's program. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown:

Constraint Television Radio Online

Audience per advertisement $110,00017,00030,000$

Cost per advertisement $$2,500$ $$300$ $$500$

Maximum media usage $102310$

To ensure a balanced use of advertising media, radio advertisements must not exceed $40\%$ of the total number of advertisements authorized. In addition, television should account for at least $10\%$ of the total number of advertisements authorized.

$($a$)$ If the promotional budget is limited to $$20,500,$ how many commercial messages should be run on each medium to maximize total audience contact? If your answer is z