Question

A swimming club can currently accommodate up to 100 members for a year-long training program. They plan to increase their pool capacity to accommodate up to an additional 75 members into the program every year. To make a profit, the club must have at least 50 members enrolled in the program. The club manager expects the minimum membership requirement for making a profit to increase by 20% every year. The manager modeled this situation with a system of inequalities, where x is time from today, in years, and y is the number of members. She graphed the system to show the solution region when the club is making a profit. What system of inequalities did the manager create? Option A:

y≥100+75x
y≥50(1.2) x

Option B:
y≥100+75x
y≥50(1.2) 2x

​Option C:
y≥100+75x
y≥50(1.2) 0.2x

Option D:
y≥100+75x
y≥50(1.2) 0.2x

​

Answer 1

The manager's scenario indicates a linear increase in club capacity and an exponential increase in the minimum members needed for profit, leading to a system of inequalities Option C: y ≥ 100 + 75x, y ≥ 50(1.2)^x.

Step-by-step explanation:

The system of inequalities the manager created represents the conditions for the swimming club's membership and profit requirements over time. This would be shown in two inequalities:

1. The first inequality accounts for the maximum number of members the club can accommodate, which initially is 100 members but increases by 75 each year (x represents the number of years). In this case, the number of members (y) must be greater than or equal to this changing capacity, which could be written as y ≥ 100 + 75x.
2. The second inequality reflects the minimum number of members required for the club to make a profit, which starts at 50 and increases by 20% each year. The form of this inequality has to account for the exponential growth, which is represented as y ≥ 50(1.2)^x.

When these two criteria are combined, and taking into consideration the growth expectation for both capacity and profit requirement, the system of inequalities would likely resemble either Option A or Option C, as B and D involve an exponent of 2x which does not align with the manager's description. However, Option A has an error that does not abide by mathematical notation (i.e., the use of consecutive variables without an operator), which makes Option C: y ≥ 100 + 75x, y ≥ 50(1.2)^x the most plausible model provided the options given.