MCD4500 - Engineering Mathematics Assignment 2 - Monash College

Assignment Task

3. A tank contains a mixture of salt and water. At time t minutes, let S(t) be the amount of salt in in the tank, let V(t) be the volume of water in Monatration of the saltwater mixture in the tank. in the tank, and let C(t) 023 s be

When a new supply of saltwater with a concentration of c grams of salt per litre is poured into the tank at rate of r1 litres per minute, mixed, and drained at rate of modelled by the ordinary differential equation s(t) = c. r1 - c(t).r2

(a) Suppose that the tank initially contain 10 litres of water, and that r1 = 16 and r2 = 12. Determine an expression for V(t).

(b) Further suppose that the tank initially contains S(o)  grams of salt and that c= 5. Determine an expression for S(t) for all t>0.

(c) Show that C(t) always approaches the same value as t→ ∞, regardless of the value of S0

4. Consider the ordinary different equation ege nsider the ordinary f"(t)-7f'(t)+12f(t) = e4t subject to the initial condition f = 0 and f'(0) = 0.

(a) Calculate the Laplace transform of f.  Expreess your final answer with a factorised denominator.

(b) Hence, use convolution theorem to determine f.

Note: State each Laplace trasform property as you use it. Refer to each property using its row number in the Table of Laplace Transforms provided. For example, L{1}= 1/8 by LT1.

Mathematical Communication: You will be assessed on your mathematical communication. Read the student Writing Guide for assistance with this. You should attempt to implement good mathematical communication in all of your work. This includes:

• Clear and coherent sentences including a concluding sentence which should of Cole Working in the context of the question;
• Clear explanations and reasoning for key steps in your
• Correct notation during calculationtions. ge ashalle202 onash Colle lege 2023 Mo 223

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