.MCAD 303010000 1 79 51 0 .CMD FORMAT rd=d ct=10 im=i et=3 zt=15 pr=3 mass length time charge temperature tr=0 vm=0 .CMD SET ORIGIN 0 .CMD SET TOL 0.001000000000000 .CMD SET PRNCOLWIDTH 8 .CMD SET PRNPRECISION 4 .CMD PRINT_SETUP 0.500000 1.200000 0.500000 0.500000 1 .CMD HEADER_FOOTER 1 1 *empty* *empty* *empty* 0 1 *empty* *empty* *empty* .CMD HEADER_FOOTER_FONT fontID=14 family=Arial points=10 bold=0 italic=0 underline=0 .CMD HEADER_FOOTER_FONT fontID=15 family=Arial points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE_NAME fontID=0 name=Variables .CMD DEFINE_FONTSTYLE_NAME fontID=1 name=Constants .CMD DEFINE_FONTSTYLE_NAME fontID=2 name=Text .CMD DEFINE_FONTSTYLE_NAME fontID=4 name=User^1 .CMD DEFINE_FONTSTYLE_NAME fontID=5 name=User^2 .CMD DEFINE_FONTSTYLE_NAME fontID=6 name=User^3 .CMD DEFINE_FONTSTYLE_NAME fontID=7 name=User^4 .CMD DEFINE_FONTSTYLE_NAME fontID=8 name=User^5 .CMD DEFINE_FONTSTYLE_NAME fontID=9 name=User^6 .CMD DEFINE_FONTSTYLE_NAME fontID=10 name=User^7 .CMD DEFINE_FONTSTYLE fontID=0 family=Times^New^Roman points=12 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=1 family=Times^New^Roman points=12 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=2 family=Times^New^Roman points=12 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=4 family=Arial points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=5 family=Courier^New points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=6 family=System points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=7 family=Script points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=8 family=Roman points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=9 family=Modern points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=10 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 .CMD UNITS U=1 .CMD DIMENSIONS_ANALYSIS 0 0 .TXT 3 1 23 0 Cg b78.800000,78.800000,814 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\b \fs28 Second Degree Polynomial and Other Models}{\par \par }{\i \fs24 Zoran Zdravkovski, Institute of Chemistry, Skopje, Macedoni}{\i \fs24 a}{\par \par The drying time of varnish depends on the amount of a certain chemical additive: \par \par }{\i m}{/g 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 \par }{\i t}{/h 7.2 6.7 4.7 3.7 4.7 4.2 5.2 5.7 \par \par The experimental data for the drying time }{\i vs.}{ mass of additive shows a parabolic trend. Determine the }{\i a}{, }{\i b}{ and }{\i c}{ constants that would fit the formula: \par \par }{\i y = cx}{\i \fs16 \up8 2}{\i + bx + a}{\par \par This type of problem in Mathcad can be handled by the built in function }{\b \i linfit}{. The input for the experimental data are in the usual vector form: \par }} } .EQN 49 3 2 0 {0:x}NAME:({8,1}87654321) .EQN 0 10 3 0 {0:y}NAME:({8,1}5.75.24.24.73.74.76.77.2) .TXT 12 49 26 0 Cg a17.200000,17.800000,107 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {Similar functions can be defined for other power functions, or even for multiple regression analysis}} } .TXT 4 -60 24 0 Cg a42.300000,42.900000,65 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain { Then a function }{\i F(x)}{ of the following type is defined:}} } .EQN 0 46 8 0 {0:F}NAME({0:x}NAME):({3,1}1鰗0:x}NAME({0:x}NAME)^(2)) .TXT 10 -46 7 0 Cg a28.000000,73.000000,51 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {The syntax for the }{\b \i linfit}{ function is:}} } .EQN 0 46 9 0 {0:const}NAME:{0:linfit}NAME({0:x}NAME,{0:y}NAME,{0:F}NAME) .TXT 11 -46 25 0 Cg a27.100000,75.800000,38 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {The constants for the equation are:}} } .EQN 0 46 10 0 {0:const}NAME={0}?_n_u_l_l_ .TXT 19 -19 20 0 Cg a25.400000,35.000000,63 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {Regression values \par for the experimental values of }{\i x}{.}} } .EQN 1 -29 17 0 {0:i}NAME:0;{0:last}NAME({0:x}NAME) .EQN 0 13 11 0 ({0:yr}NAME)[({0:i}NAME):{0:F}NAME(({0:x}NAME)[({0:i}NAME))*{0:const}NAME .EQN 1 46 15 0 {0:yr}NAME={18994}?_n_u_l_l_ .TXT 15 -58 21 0 Cg a40.600000,109.500000,60 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i yg}{ is defined only to get a smooth regression graph:}} } .EQN 4 0 12 0 {0:r}NAME:0,0.01;8 .EQN 0 11 14 0 {0:yg}NAME({0:r}NAME):{0:F}NAME({0:r}NAME)*{0:const}NAME .EQN 2 17 16 0 &&(_n_u_l_l_&_n_u_l_l_)&{0:yg}NAME({0:r}NAME),({0:y}NAME)[({0:i}NAME),5@&&(_n_u_l_l_&_n_u_l_l_)&{0:r}NAME,({0:x}NAME)[({0:i}NAME) 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 1 0 0 NO-TRACE-STRING 1 0 1 0 NO-TRACE-STRING 0 3 2 0 NO-TRACE-STRING 0 4 3 0 NO-TRACE-STRING 0 1 4 0 NO-TRACE-STRING 0 2 5 0 NO-TRACE-STRING 0 3 6 0 NO-TRACE-STRING 0 4 0 0 NO-TRACE-STRING 0 1 1 0 NO-TRACE-STRING 0 2 2 0 NO-TRACE-STRING 0 3 3 0 NO-TRACE-STRING 0 4 4 0 NO-TRACE-STRING 0 1 5 0 NO-TRACE-STRING 0 2 6 0 NO-TRACE-STRING 0 3 0 0 NO-TRACE-STRING 0 4 1 0 NO-TRACE-STRING 0 1 19 19 0 3 .TXT 26 -27 28 0 Cg a70.400000,76.300000,91 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {Find the amount of the additive for which the drying time can be expected to be minimum.}} } .TXT 5 7 29 0 Cg a65.800000,66.000000,124 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {The first derivative of the function gives its minimum (or maximum). In this case \par y'=2cx-b or for y'=0, x=-b/2c.}} } .EQN 7 -4 34 0 {0:c}NAME:({0:const}NAME)[(0) .EQN 0 12 35 0 {0:c}NAME={0}?_n_u_l_l_ .EQN 3 -12 36 0 {0:b}NAME:({0:const}NAME)[(1) .EQN 0 12 37 0 {0:b}NAME={0}?_n_u_l_l_ .EQN 3 -12 38 0 {0:a}NAME:({0:const}NAME)[(2) .EQN 0 12 39 0 {0:a}NAME={0}?_n_u_l_l_ .EQN 7 -15 30 0 {0:xmin}NAME:-(({0:b}NAME)/(2*{0:c}NAME)) .EQN 0 24 31 0 {0:xmin}NAME={0}?_n_u_l_l_ .TXT 0 14 40 0 Cg a37.000000,39.500000,48 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {amount of varnish giving minimum drying time }} } .EQN 6 -38 32 0 {0:tmin}NAME:{0:c}NAME*(({0:xmin}NAME))^(2)+{0:b}NAME*{0:xmin}NAME+{0:a}NAME .EQN 0 24 33 0 {0:tmin}NAME={0}?_n_u_l_l_ .TXT 0 14 41 0 Cg a16.600000,35.800000,22 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {minimum drying time}} } .TXT 9 -38 50 0 Cg a57.000000,75.800000,74 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {Find the amount of additive for which the drying time would be 5 hours?}} } .EQN 4 22 42 0 {0:y}NAME:5 .EQN 3 -21 43 0 {0:x1}NAME:1 .EQN 0 34 44 0 {0:x2}NAME:10 .EQN 4 -34 45 0 {0:x1}NAME:{0:root}NAME({0:c}NAME*({0:x1}NAME)^(2)+{0:b}NAME*{0:x1}NAME+{0:a}NAME-{0:y}NAME,{0:x1}NAME) .EQN 0 34 46 0 {0:x2}NAME:{0:root}NAME({0:c}NAME*({0:x2}NAME)^(2)+{0:b}NAME*{0:x2}NAME+{0:a}NAME-{0:y}NAME,{0:x2}NAME) .EQN 4 -34 47 0 {0:x1}NAME={0}?_n_u_l_l_ .EQN 0 34 48 0 {0:x2}NAME={0}?_n_u_l_l_ .TXT 4 -34 49 0 Cg a73.700000,79.100000,100 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {Since this is a quadratic equation, there are two solutions. They are both real (see the graph).}} } .TXT 7 -3 51 0 Cg a70.300000,71.000000,418 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\b References}{:\par \par 1. Z. Zdravkovski, Mathcad in Chemistry Calculations, }{\i J. Chem. Ed.}{ }{\b 1991}{, }{\i 68}{, A95 and }{\b 1992}{, }{\i 69}{, A240.\par \par 2. T. R. Dickson, The Computer and Chemistry, W. H. Freeman, San Francisco, 1968 .\par \par 3. P. W. Atkins, Physical Chemistry, W. H. Freeman, New York, 4th Ed., 1990.\par \par 4. J Freund, Modern Elementary Statistics, Prentice-Hall, Englewood Cliffs, 1979.}} }