1. A gym has 200 members who pay $30 per month for unlimited use of the gym's equipment. A survey of the members indicates that for each $5 increase in the monthly fee, the gym will lose 20 members. This means that the revenue R from fees, which is currently $6000 per month, will become, where f is a whole number of $5 fee increases. Write and solve a quadratic inequality to answer the questions: For what numbers of $5 fee increases will the revenue from fees actually be less than its current value?

200 members pay $30 each

Current revenue = 200 x $30

= $6000

For each $5 increase in fee, 20 members will leave

For i number of $5 increase, 20i members will leave

New number of members = 200 - 20i

For i number of $5 increase, new fee = 30 + 5i

Therefore, new revenue = (200 - 20i) x (30 + 5i)

= 200 (30 + 5i) - 20f (30 + 5i)

= 6000 + 1000i - 600i - 100i²

= 6000 + 400i - 100i²

In the form of a quadratic equation,

-100i² + 400i + 6000

R (i) = - 100i² + 400i + 6000

This is not our answer yet.

We are told to write a quadratic inequality for what numbers of $5 fee increases will the revenue from fees actually be less than its current value

Current value = 6000

Therefore, we are looking for i, when R (i) < 6000

Remember, R (i) = - 100i² + 400i + 6000

Therefore,

R (i) < 6000 = - 100i² + 400i + 6000 < 6000

We are to solve the quadratic inequality,

-100i² + 400i + 6000 < 6000

⇒ - 100i² + 400i < 0 (Subtract 6000 from both sides)

⇒ 400i < 100i² (Add 100i to both sides)

⇒ 4i < i² (Divide both sides by 100)

⇒ 4 < i (Divide both sides by i)

4 4

Therefore, for more than 4 numbers of $5 increase in the fees, the revenue from fees will actually be less than its current value.