Identify the given differential equation and parameters.

Step 1: Identify the given differential equation and parameters. The rate of change of mass y(t) of RA_2 is given by the differential equation: (dy)/(dt) = 50e^-10t - ky We are given k = 2 kg/sec and the initial condition y(0) = 40 kg. For the equation to be dimensionally consistent, if y is in kg and dy/dt is in kg/sec, then k must be in 1/sec. We will assume that the unit "kg/sec" for k is a typo and use k=2 with units of 1/sec. Step 2: Substitute the value of k into the differential equation. (dy)/(dt) = 50e^-10t - 2y Step 3: Rearrange the equation into the standard form of a first-order linear differential equation, (dy)/(dt) + P(t)y = Q(t). (dy)/(dt) + 2y = 50e^-10t Here, P(t) = 2 and Q(t) = 50e^-10t. Step 4: Calculate the integrating factor, I(t) = e^ P(t) dt. I(t) = e^ 2 dt = e^2t Step 5: Multiply the differential equation by the integrating factor. e^2t ( (dy)/(dt) + 2y ) = e^2t (50e^-10t) The left side is the derivative of the product y(t)I(t): (d)/(dt)(y e^2t). The right side simplifies to 50e^2t - 10t = 50e^-8t. So, the equation becomes: (d)/(dt)(y e^2t) = 50e^-8t Step 6: Integrate both sides with respect to t. (d)/(dt)(y e^2t) dt = 50e^-8t dt y e^2t = 50 ( -(1)/(8) e^-8t ) + C y e^2t = -(25)/(4) e^-8t + C Step 7: Solve for y(t) by dividing by e^2t. y(t) = e^-2t ( -(25)/(4) e^-8t + C ) y(t) = -(25)/(4) e^-10t + C e^-2t Step 8: Use the initial condition y(0) = 40 kg to find the constant C. Substitute t=0 and y(0)=40: 40 = -(25)/(4) e^-10(0) + C e^-2(0) 40 = -(25)/(4) (1) + C (1) 40 = -(25)/(4) + C C = 40 + (25)/(4) C = (160)/(4) + (25)/(4) C = (185)/(4) Step 9: Substitute the value of C back into the equation for y(t). y(t) = -(25)/(4) e^-10t + (185)/(4) e^-2t The mass y(t) of RA_2 for t 0 is: y(t) = (185)/(4) e^-2t - (25)/(4) e^-10t kg