Please show every step used to solve this Modeling A baked potato, which has bee

Question

Please show every step used to solve this

Modeling A baked potato, which has been marinating in the refrigerator at 40degree, is placed on a grill that has temperature 500degree. After 3 minutes, the potato is at 70degree. How long will it take for the potato to reach 130degree? Lake Euler has 2 rivers flowing into it and 1 river flowing out. The 2 inflow rivers are the Newton River (flow rate r1 = 3 times 105 m3/day, pollution concentration C1 = 10-3 kg/m3) and the Leibniz River (flow rate r2 = 2 times 105 m3/day, pollution concentration C2 = 5 times 10-3 kg/m3). The pollution in Lake Euler is well-mixed, and the outflow river, the Akron River, has a flow rate of r3 = 5 times 105 m3/day so the volume is constant at 108 m3. Set up and solve the differential equation for the mass of pollution in Lake Euler as a function of time, given that the initial concentration is 5 times 10- 3 kg/m3.

Explanation / Answer

1. The first question is about heat content of a body and it can be thought of as a first order process

if Q(t) represents the heat content of the potato at time t

then we can write dQ(t) / dt = k Q(t) which means that the rate of change in heat content of body is proportional to the value of heat content of th body., k is a parameter of the process

now multiplying both sides with dt

d Q(t) / Q(t) = dt

Integrating and rearranging you will get

Q(t) = C e(kt) where C is a constant

Now at t= 0, Q(0) = 40

at t= 180 sec, Q(180) = 70

first put t = 0

Q(0) = C => C = 40

So our equation is Q(t) = 40 e(kt)

now put t= 180 ; Q(180) = 70

70 = 40 e(180k)

or k = 3.1089 * 10^-3

when Q(t) = 130, then 40 e(kt) = 130

=> e(kt) = 3.25

or t = 379.12 seconds or 6 min 19 seconds

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