## Thursday, February 13, 2020

### Analyse and model engineering situations and solve problems using Math Problem - 1

Analyse and model engineering situations and solve problems using Ordinary differential equations - Math Problem Example This empirical observation coincides to the NewtonÃ¢â‚¬â„¢s Law of Cooling which states that Ã¢â‚¬Å"the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature.Ã¢â‚¬  For instance, when a hot metal ball is placed in a bath of tap water at temperature of T0, it gradually cools. In this process which takes time to complete, naturally the metal ball gives off heat to the surrounding water so that the bath gets warm due to the heat released to it. However, as time proceeds, since the bath of water is open to the larger environment at T0, the system consisting of it and the material it contains would in time establish equilibrium with its environment. In which case, the NewtonÃ¢â‚¬â„¢s Law of Cooling applies such that, for the heated object being cooled within a room, the temperature of the hot body changes so that it approaches the roomÃ¢â‚¬â„¢s temperature which is T0. b. Formulate mathematical model for t he cooling process. According to NewtonÃ¢â‚¬â„¢s Law of Cooling with which the problem statement proves to be consistent, a first-order differential equation may be set up as follows: = -kT where Ã¢â‚¬ËœkÃ¢â‚¬â„¢ refers to the constant of proportionality. The negative sign accounts for the difference in temperature since the object being cooled would have a lower final temperature compared to its initial temperature. Then on solving the equation: = -k Where Tf = final temperature difference T(t) - T0 Ti = initial temperature difference T1 - T0 Here, T0 = ambient room temperature T1 = initial temperature of heated object T(t) = temperature of the object (under cooling) at anytime Ã¢â‚¬ËœtÃ¢â‚¬â„¢ So that upon evaluation of the integral, ln Tf - ln Ti = -kt By exponent property, ln = -kt ---? = ---? = Then, substituting expressions for Tf and Ti: ----? T(t) - T0 = (T1 - T0) 2. At time t = 0 water begins to leak from a tank of constant cross-sectional area A. The rate of outflow is pro portional to h, the depth of water in the tank at time t. Write the constant of proportion kA where k is constant. a. Analyse the tank leaking process. Since water leaks out of the tank from an initial height say h0 which corresponds to water volume of V(h0) in the tank, the finite change in this volume per unit change in time, beginning at t = 0 would be (?V/t). The tank is not being filled in so this merely represents the rate of water outflow which is proportional to the water depth in the tank. Essentially, the water depth may be expressed as the finite change in height h(t) - h0 as the water leaks out of the tank where h0 refers to the initial height in the tank and h(t) is the height of the water measured at any time Ã¢â‚¬ËœtÃ¢â‚¬â„¢. b. Formulate mathematical model for the leaking process. The leaking process may be mathematically modelled as follows: = - k A ?h in which A pertains to the constant area of cross-section through water depth The factors kA serve as the constant of proportionality and the negative sign is used to signify the value of h(t) that is lower than h0. For depth ?h = h(t) - h0, it follows that ?V = V[h(t)] - V(h0). Thus, = -k A [ h(t) - h0 ] which on arranging yields to: Write conclusions based on your formulated mathematical model for leaking process. Task 2 Ã¢â‚¬â€œ Learning Outcome 4.2 Solve first order differential equations using analytical and numerical methods. 3. Find the solution of the following equations: a. Separating variables, = t dt Integrating both sides, let u