Write an equation for V, the amount of coffee remaining in the pot at any time t.

Question

The rate at which the amount of coffee in a coffee pot changes with time is given by the differential equation dV / dt = kV where V is the amount of coffee left in the coffee pot at any time t seconds. At time t = 0 There are 16 oz of coffee in the coffee pot at time t = 80 there were 8 oz of coffee.

a) Write an equation for V, the amount of coffee remaining in the pot at any time t.

b) At what rate is the amount of coffee in the pot decreasing when there are 4 ounces of coffee remaining.

c) At what time t will the have 2 ounces of coffee remaining.

Answer 1

a)

dV/dt = KV

dV/V = Kdt

Integrating both sides

ln(V) = Kt + C₁

V = C₂e^Kt

Solve for C₂ and K

V(0) = 16 = C₂

V(80) = 8 = 16e^80K

8/16 = 1/2 = e^80K

80K = ln(1/2) = -ln(2)

K=-ln(2)/80

V(t) = 16e^-ln(2)t/80 = 16(2)^-t/80

b) dV/dt = KV = (-ln(2)/80)(4) = -ln(2)/20 (Negative because it is decreasing)

c) 2 =16(2)^-t/80

1/8 = 2^-t/80

-ln(8) = -t/80ln(2)

t = -80ln(8)/-ln(2) = 80(3) = 240 seconds = 4 minutes