A gambling casino is designed in the shape of a square pyramid with a side length of 75 ft. The height of the building is 28 ft. An air filtering system can circulate air at 26 comma 250 cu ft/hr. How long will it take the system to filter all the air in the room?

Step-by-step explanation:

Given the dimension of a square pyramid

Length of pyramid

L = 75 ft

Height of pyramid

H = 28ft

The pyramid is a square base

Then,

Volume of a pyramid is calculated using

Volume = base area * height

Since base is a square, then the area of a square is

Area = L² = 75² = 5625 ft²

Then,

Volume = base Area * height

Volume = 5625 * 28

Volume = 157,500 ft³

Then, given the rate of volume to be

dV / dt = 26,250 ft³ / hr

So, we want to find the time the system will filled the air

dV / dt = 26,250

dV = 26,250 dt

Integrate both sides

∫ dV = ∫ 26,250 dt

V = 26,250t + C

When t = 0, V = 0, initially, there is not volume of system air.

Therefore C = 0

V = 26,250t

When V = 157,500 ft³

Then,

t = V / 26,250

t = 157,500 / 26,250

t = 6 hours

So, it will take 6 hours for the system to circulate the room