Answer to Question #265375 in Differential Equations for joanne

Question #265375
  1. A certain substance was placed inside a room where the temperature is 17°C. it is observed that after 30 seconds, the temperature of the substance drops to 27°C and after 1 minute, the temperature drops to 20°C. what is the initial temperature of the body? Ans. 50.33°C
  2. The bureau of census record in 1975 shows that the population in the country doubles compared to that of 1955. In what year will the population quadruple? Ans. 1995
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Expert's answer
2021-11-15T16:07:33-0500

\frac{}{} The change in temperature is given by dθdt=k(17θ)    dθ17θ=kdt    θ=17cekt(1)at t = 60 seconds and θ=20, we have that3=ce60k(3)at t = 30seconds and θ=27, we have thatDividing (3) by (2), we have0.3=e30k    k=0.0401Putting k =0.0401 in 3 we have thatc=33.33Next, we substitute k and c in (1), at time 0 to get our solutionθ=17+33.33e0=50.330Cdpdt=kt    dpp=kt    lnp=kt+A    p=CektAt t = 20, p=2C, as given in the question    2=e20k    ln2=20k    k=0.0347When p=4C, we have that4=e0.0347t    t=39.95 approximately 40 years. Hence p quadruples at1955 + 40years=1995\text{The change in temperature is given by }\\ \frac{d\theta}{dt}= k(17-\theta)\\ \implies \frac{d\theta}{17-\theta}=kdt\\ \implies \theta = 17-ce^{-kt}-(1)\\ \text{at t = 60 seconds and $\theta = 20$, we have that}\\ -3=ce^{-60k}-(3)\\ \text{at t = 30seconds and $\theta = 27$, we have that}\\ \text{Dividing (3) by (2), we have}\\ 0.3 = e^{-30k}\\ \implies k = 0.0401\\ \text{Putting k =0.0401 in 3 we have that}\\ c = -33.33\\ \text{Next, we substitute k and c in (1), at time 0 to get our solution}\\ \therefore \theta = 17+33.33e^0=50.33^0C\\ \frac{dp}{dt}=kt\\ \implies \frac{dp}{p}=kt\\ \implies \ln p =kt +A\\ \implies p = Ce^{kt}\\ \text{At t = 20, p=2C, as given in the question}\\ \implies 2 = e^{20k}\\ \implies \ln 2 = 20k\\ \implies k = 0.0347\\ \text{When p=4C, we have that}\\ 4 = e^{0.0347t}\\ \implies t = 39.95 \text{ approximately 40 years. Hence p quadruples at}\\ \text{1955 + 40years=1995}


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