# math

In a room of 20.000 ft^3 of air, 600 ft^3 of fresh air flows in per minute, and the mixture (made practically inform by circulating fans) is exhausted at a rate of 600 cubic feet per minute (cfm). What is the amount of fresh air inside the room y(t) at any time, if y(0)=0? After what time will 90% of air in the room be fresh.

1. 👍 0
2. 👎 0
3. 👁 753
1. I assume you are familiar with integration or differential equations.

y(t)=amount of air (in c.f.) in room at time t.

Then
dy/dt
=y'(t)
=rate of increase of fresh air per unit time (min)
=fresh air in - fresh air out per minute
=600 - 600(y/20000)
=600*(20000-y)/20000
Separate variables and integrate:
∫dy/(20000-y) = (600/20000)∫ dt
-log(20000-y) = 600t/20000 + C'
log(20000-y) = -600t/20000 + C
Raise to power of e:
20000-y = e^(-600t/20000+C)
y=20000(1-e^(-600t/20000+C)
At t=0, y=0
=>
0=(1-e^(0+C))
=> C=0
Therefore:
y=20000(1-e^(-600t/20000))
at y=0.9*20000=18000,
=>
e^(-600t/20000)=0.1

t=-(20000/600)*log(0.1)
=77 minutes (approx.)

1. 👍 0
2. 👎 0

1. 👍 0
2. 👎 0
3. You're welcome!

1. 👍 0
2. 👎 0
4. Te amo :3

1. 👍 0
2. 👎 0

## Similar Questions

1. ### Physics

A room has a volume of 120 m^3. An air-conditioning system is used to replace the air in the room every twenty minutes, using ducts that have a square cross ection. Assuming that air can be treated as an incompressible fluid,

asked by ALan on March 16, 2010
2. ### engineering

An aircraft flies at an altitude of 30,000 feet. Determine the air temperature (in [K]), air pressure (in [Pa]) and air density (in [kg/m3]) at this altitude, according to the standard atmosphere.

asked by sam on July 7, 2017
3. ### Chemistry

The density of air at ordinary atmospheric pressure and 25°C is 1.19 g/L. What is the mass, in kilograms, of the air in a room that measures 12.0 multiplied by 15.5 multiplied by 8.4 ft?

asked by Anonymous on August 18, 2013
4. ### Chemistry

A student breaks a thermometer and spills most of the mercury (Hg) onto the floor of the laboratory that measure 15.2 m long, 6.6 m wide and 2.4 m high. a) (1 mark) Calculate the volume of the room in m3 and L. b) (2 marks) If the

asked by Anonymous on December 13, 2017
5. ### accounting

Fill in the blanks in the following separate income statements a through e. Identify any negative amount by putting it in parentheses. a b c d e Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \$60,000

asked by Joann on July 9, 2011
1. ### College Physics

A 1500-W heater is designed to be plugged into a 120-V outlet. What current will flow through the heating coil when the heater is plugged in? I = A What is R , the resistance of the heater? How long does it take to raise the

asked by Lanise on February 6, 2010
2. ### chemistry

an average adult inhales a volume of 0.50 liter of air into the lungs with each breath.If the air is warmed from room temperature (20 degree celsius) to body temperature (37 degree celsius),what is the volume of air exhaled?

asked by MHECY on January 19, 2016
3. ### physics help

A cylindrical air duct in an air conditioning system has a length of 3.5 m and a radius of 5.00 10-2 m. A fan forces air (ç = 1.8 1 x0-5 Pa·s) through the duct, such that the air in a room (volume = 242 m3) is replenished every

asked by peter on December 4, 2010
4. ### Chemistry

When you try to leave your room you learn that the only door out is locked. Assuming there is no air flow into the room (air tight with no ventilation system) how long can you survive in the room (5 meters long, 5 meters wide, 5

asked by Sarah on November 13, 2016
5. ### physics

A room has the dimensions 3.0 m by 4.0 m by 4.0 m . The air in the room is 79% N2( MW= 28) and 21% O2 (MW = 32). At a temperature of 20 degrees Celsius and pressure of 115 kPa , what is the mass of the air in the room in

asked by ED on November 28, 2011