The problem as posted is not very clear. I think it is your interpretation of the original problem as shown in a book or somewhere else.Originally Posted byusm_67

----What is x? Is that number of rooms filled? So if price of rent per room remains at $20, then x=30? If price is $21, x=29?

----To clean the room? To clean one room only? Or should that be "to clean the rooms, the rooms that are filled? The x-rooms?

Let me solve the problem based on the following, per day:

---x = number of rooms filled.

---C = x^2 -60x +900 is cost for all filled rooms, for the x-rooms.

Here is one way.

Break even. Meaning, revenue = cost. No loss, no profit.

Revenue = x*(rent of a room) ---***

Rent of a room = $20 +$1*(30-x) = 20 +30 -x = (50-x) dollars.

where (30-x) is the number of rooms emptied. [See what the "x" did, too much confusion.]

See what $(50-x) means:

If x=30, or no empty room, (50-x)=(50-30)=$20 per room.

If x=29, (50-x)=(50-29)=$21 per room.

It is what the problem say.

So, Revenue, R = x*(50-x) = (50x -x^2) dollars.

Equate that to the C = x^2 -60x +900 for break even,

x^2 -60x +900 = 50x -x^2

x^2 -60x +900 -50x +x^2 = 0

2x^2 -110x +900 = 0

x^2 -55x +450 = 0

Use the Quadratic Formula,

x = {-(-55) +,-sqrt[(55)^2 -4(1)(450)]} /(2*1)

x = (55 +,-35)/2

x = 45 or 10 rooms filled.

Of course, 45 rooms filled is impossible---the motel has max 30 rooms only.

Hence, x = 10 rooms filled.

Therefore, for break even, 10 rooms must be filled up, or 20 rooms must be vacant. -----answer.