Science:Math Exam Resources/Courses/MATH101/April 2009/Question 08 (a)
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Question 08 (a) 

Consider the chemical reaction Suppose at time t = 0 sec the concentration of chemical A is 0.1 mol/L, the concentration of chemical B is 0.2 mol/L, and the concentrations of chemicals C and D are both 0. For t ≥ 0, let x(t) be the concentration of chemical D in mol/L. It can be shown that x(t) is the solution to the initialvalue problem where k is a positive constant whole value can be determined by experiment.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Start off by cross multiplying. Then integrate (you may want to use partial fractions to integrate). I hope you ate your Wheaties because this is a long one... 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Cross multiplying our equation, we have that Now we integrate. The right hand side integrates to As for the left hand side, we integrate by partial fractions. Now we can either compare coefficients or plug in values for x. I will choose to plug in values for x. The above tells us that when , we have that and so . When , we have and so . Hence, we have (technically the above should add a constant but we can include this in a minute.) Hence becomes Using the initial condition of , we see that our constant is and simplifying, this becomes Our equation simplifies to Exponentiating the far left and the far right hand sides of the above gives Taking the tenth root of both sides yields Removing the absolute values, we have Plugging in the initial condition of , we see that only the positive value above works. Hence Now, we cross multiply to see that Bringing the values to one side gives Lastly, isolating for gives completing the question. 