Solved: Chemical Reactions Chemicals A and B combine in a 3-to-1 ratio to form a

Chapter 5, Problem 95

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Chemicals A and B combine in a 3 -to- 1 ratio to form a compound. The amount of compound x being produced at any time t is proportional to the unchanged amounts of A and B remaining in the solution. So, when 3 kilograms of A is mixed with 2 kilograms of B, you have

\(\frac{d x}{d t}=k\left(3-\frac{3 x}{4}\right)\left(2-\frac{x}{4}\right)=\frac{3 k}{16}\left(x^{2}-12 x+32\right)\)

One kilogram of the compound is formed after 10 minutes. Find the amount formed after 20 minutes by solving the equation

\(\int \frac{3 k}{16} d t=\int \frac{d x}{x^{2}-12 x+32}\)

Text Transcription:

frac d x d t=k(3-frac 3 x 4)(2-frac x 4)=frac 3 k 16(x^2-12 x+32)

int frac 3 k 16 d t=int frac d x x^2-12 x+32