A 17650 cubic foot room initially has a radon level of 870 picocuries per cubic foot. A ventilation system is installed that brings in 385 cubic feet of air per hour that contains 5 picocuries per cubic foot, while an equal quantity of the well-mixed air in the room leaves the room each hour. Setup and use a differential equation to determine how long it will take for the room to reach a safe to breathe level of 105 picocuries per cubic foot. (Round your answer to 5 decimal places.)
N(0) = 870 *17650
d N/dt = 385 *5 - 385 (N/17650)
dN/dt = 385[5 - (N/17650)]
when does N = 17650(105) ?
say dx/dt = a - k x
try
x = a/k -c e^-kt
here
dN/dt = 1925 - .0218 N
so
k = .0218
a = 1925
and N = 88250 - c e^-.0218 t
when t = 0
e^-.0218 t = 1, use to get c
find when N = 105*17650
CHECK MY ARITHMETIC STEP BY STEP.
To solve this problem, let's define some variables:
Let R(t) be the radon level in picocuries per cubic foot at time t.
Let V(t) be the volume of radon in the room in picocuries at time t.
Given Information:
Initial radon level: R(0) = 870 picocuries per cubic foot
Rate of air flow: Inflow = 385 cubic feet per hour, Radon concentration in inflow = 5 picocuries per cubic foot
Radon concentration leaving the room: Radon concentration in outflow = R(t) picocuries per cubic foot
Rate of air leaving the room: Outflow = 385 cubic feet per hour
The rate of change of the radon level is equal to the rate of change of the volume of radon in the room divided by the volume of the room.
dR/dt = (dV/dt) / (17650)
To find the rate of change of the volume of radon in the room, we need to consider the inflow and outflow.
The rate of change of the volume of radon in the room is equal to the rate of radon inflow minus the rate of radon outflow.
dV/dt = (Rate of Radon Inflow) - (Rate of Radon Outflow)
The rate of radon inflow is given by:
Rate of Radon Inflow = (Inflow) * (Radon concentration in inflow) = (385) * (5)
The rate of radon outflow is given by:
Rate of Radon Outflow = (Outflow) * (Radon concentration in outflow) = (385) * (R(t))
Now, substitute these values into the differential equation:
dV/dt = (385) * (5) - (385) * (R(t))
Finally, divide the differential equation by the volume of the room (17650) to find the differential equation in terms of R(t):
(dR/dt) = [(385) * (5) - (385) * (R(t))] / (17650)
We can solve this differential equation to find how long it will take for the room to reach a safe to breathe level of 105 picocuries per cubic foot.
To setup a differential equation for this problem, we need to consider the rate of change of radon level with respect to time.
Let's denote the rate of radon change in the room as dR/dt (picocuries per cubic foot per hour), and the radon level in the room as R (picocuries per cubic foot).
We know that the ventilation system brings in air at a rate of 385 cubic feet per hour, and this air has a radon level of 5 picocuries per cubic foot. So the rate at which radon is entering the room is 385 * 5 = 1925 picocuries per hour.
We also know that an equal quantity of the well-mixed air in the room leaves the room each hour. Since the room has a volume of 17650 cubic feet, the rate at which radon is leaving the room is (R/17650) * 17650 = R picocuries per hour.
Therefore, the overall rate of change of radon level in the room can be expressed as:
dR/dt = 1925 - (R/17650) * 17650
Now, we want to find how long it will take for the room to reach the safe level of 105 picocuries per cubic foot. This happens when R = 105. So we can rewrite the differential equation as:
dR/dt = 1925 - (105/17650) * 17650
Simplifying further:
dR/dt = 1925 - 105
To solve this differential equation, we can integrate both sides with respect to t:
∫ dR = ∫ (1925 - 105) dt
Integrating, we get:
R = 1925t - 105t + C
Now, we need to find the value of C. We know that at t = 0 (initially), the radon level is 870 picocuries per cubic foot. Substituting this into the equation:
870 = 1925*0 - 105*0 + C
870 = C
Therefore, the equation becomes:
R = 1925t - 105t + 870
To find how long it will take for the room to reach a safe level of 105 picocuries per cubic foot, we need to solve for t when R = 105:
105 = 1925t - 105t + 870
Simplifying:
200t = 105 - 870
200t = -765
t = -765/200 ≈ -3.825
Since time cannot be negative, we can conclude that it will take approximately 3.825 hours for the room to reach a safe to breathe level of 105 picocuries per cubic foot.