gibson:teaching:spring-2015:math445:hw2
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gibson:teaching:spring-2015:math445:hw2 [2015/02/15 18:44] gibson |
gibson:teaching:spring-2015:math445:hw2 [2015/02/16 06:41] (current) gibson |
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| **Problem 8:** Make a histogram of 1000 random numbers from a normal (Gaussian) distribution. Use Matlab's ''hist'' and ''randn'' functions. | **Problem 8:** Make a histogram of 1000 random numbers from a normal (Gaussian) distribution. Use Matlab's ''hist'' and ''randn'' functions. | ||
| - | **Problem 9:** Chemicals A and B combine in reaction to form chemical C. The rate of production of C is proportional to the amount of A and the amount of C present at any given time: $dC/dt = \alpha A(t) B(t)$. 2 gm of A combine with 1 gm of B to form 3 gm of C. At the start of the reaction there are 40 gm A, 50 gm B, and 0 gm C. After 3 minutes there are 10 gm C. | + | **Problem 9:** Chemicals A and B combine in reaction to form chemical C. The rate of production of C is proportional to the amount of A and the amount of C present at any given time: $dC/dt = \alpha A(t) B(t)$. 2 gm of A combine with 1 gm of B to form 3 gm of C. At the start of the reaction there are 40 gm A, 50 gm B, and 0 gm C. After 5 minutes there are 10 gm of C. |
| - | You can use differential equations to solve this problem and determine the amounts of | + | This is a problem in differential equations. If you take MATH 527 you will learn how to solve this problem and determine the amounts of A, B, and C (in grams) as a function of time t (in minutes). These functions are |
| - | A, B, and C as a function of time t (measured in minutes). These functions are | + | |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | C(t) &= 300 (e^{0.0378 t}-1)/(5e^{0.0378 t} - 2) | + | C(t) &= 300 (e^{0.0227 t}-1)/(5e^{0.0227 t} - 2) |
| \end{eqnarray*} | \end{eqnarray*} | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | A(t) &= 40 - 2/3 \; A(t) | + | A(t) &= 40 - 2/3 \; C(t) |
| \end{eqnarray*} | \end{eqnarray*} | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | B(t) &= 50 - 1/3 \; A(t) | + | B(t) &= 50 - 1/3 \; C(t) |
| \end{eqnarray*} | \end{eqnarray*} | ||
| Make a plot of A, B, and C versus t for t between 0 and two hours. Label the horizontal axis and indicate which curve is which with a legend. | Make a plot of A, B, and C versus t for t between 0 and two hours. Label the horizontal axis and indicate which curve is which with a legend. | ||
| - | **Problem 10:** Use the graph to answer the following questions. You might need to refine your plot and zoom in answer some of these. | + | **Problem 10:** Use the graph from problem 9 to answer the following questions about the reaction. You might need to refine your plot and zoom in answer some of these. |
| - | (a) How much C is eventually produced?\\ | + | (a) How much A,B, and C are present after one hour?\\ |
| - | (b) Are A and B fully depleted? If not, how much of each are left at the end?\\ | + | (b) How much C is eventually produced?\\ |
| - | %%(c)%% Does your answer to (b) make sense, given the problem description? Explain. | + | %%(c)%% Are A and B fully depleted in the long-term limit? If not, how much of each are left?\\ |
| - | (d) How long does it take for the reaction to reach 99% completion? | + | (d) Does your answer to %%(c)%% make sense, given the problem description? Explain.\\ |
| + | (e) How long does it take for the reaction to reach 95% completion? | ||
gibson/teaching/spring-2015/math445/hw2.1424054647.txt.gz · Last modified: 2015/02/15 18:44 by gibson
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