a. To find the amount of pollution in the lake after 14 hours, we need to calculate the total pollution that has entered the lake and subtract the pollution that has been removed by the filter.
Total pollution entering the lake after 14 hours:
∫[0 to 14] 12(1-e^(-0.5t)) dt
= -24e^(-7) + 24
Pollution removed by the filter after 14 hours:
∫[0 to 14] 0.6t dt
= 0.6(7^2)
= 29.4
Amount of pollution in the lake after 14 hours:
-24e^(-7) + 24 - 29.4
≈ 4.854 gallons
b. To find the time when the rate of pollution entering the lake equals the rate of pollution removal, we need to set f(t) equal to g(t) and solve for t:
12(1-e^-0.5t) = 0.6t
12 - 12e^-0.5t = 0.6t
12 = 13.2e^-0.5t
e^-0.5t = 12/13.2
e^-0.5t = 0.9091
-0.5t = ln(0.9091)
t ≈ 0.191 hours
c. Amount of pollution in the lake at t = 0.191 hours:
12(1-e^(-0.5*0.191))
≈ 6.905 gallons
d. To find the time when all the pollution has been removed from the lake, we need to set the total pollution removed equal to the total pollution entered and solve for t:
∫[0 to t] 12(1-e^-0.5x) dx = ∫[0 to t] 0.6x dx
-24e^(-0.5t) + 24t = 0.3t^2
-24e^(-0.5t) = 0.3t^2 - 24t
Dividing by -24:
e^(-0.5t) = -0.0125t^2 + t
This equation does not have a simple analytical solution to find t. We can approximate the time using numerical methods.
Pollution begins to enter a lake at time t=0 at a rate (in gallons per hour) given by the formula f(t), where t is the time (in hours). At the same time, a pollution filter begins to remove the pollution at a rate g(t) as long as the pollution remains in the lake. f(t)=12(1-e^-0.5t), g(t)=0.6t
a. how much pollution is in the lake after 14 hours?
b. find the time after t=0 when the rate that pollution enters the lake equals the rate the pollution is removed.
c. Find the amount of pollution in the lake at the time found in part b
d. Find the time when all the pollution has been removed from the lake
1 answer