Final answer:
The greatest number of fans that can be in attendance at an arena with a total capacity of 20,000, including ushers, and with a requirement of one usher for every 30 fans, is 19,354.
Step-by-step explanation:
The question is asking to find the greatest number of fans that can be in attendance at a sports arena given a specific capacity and usher-to-fan ratio. The total capacity of the arena is 20,000, which includes both fans and ushers.
To find the maximum number of fans, we need to understand the usher requirement: one usher is needed for every 30 fans. We can set up an equation where f is the number of fans and u is the number of ushers. Since each usher can serve 30 fans, u = f/30. The capacity C is the sum of fans and ushers: C = f + u. Plugging in the usher equation gives us 20,000 = f + (f/30).
Multiplying both sides of the equation by 30 to clear the fraction, we have 600,000 = 30f + f or 600,000 = 31f. Dividing both sides by 31 gives us f = 19,354 (rounded down to the nearest whole number, since we can't have a fraction of a fan).
Therefore, the greatest number of fans that can be in attendance at the arena is 19,354.