document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=662256>Question 662256</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Im trying to help my sister with her hw yet i dont kno how to do this, would someone help me so i could teach her<BR>\n" );
document.write( "Milk Problem: Handy Andy sells one-gallon cartons of milk (4 quarts) for $3.09 each and half-gallon cartons for $1.65 each. Assume that the number of cents you pay for a carton of milk varies linearly with the number of quarts the carton holds. <BR>\n" );
document.write( "a) Write the particular equation expressing price in terms of quarts.<BR>\n" );
document.write( "b) If Handy Andy sold 3-gallon cartons, what would your equation predict the price to be?<BR>\n" );
document.write( "c) The actual prices for pint cartons (1/2 quart) and one-quart cartons are $.57 and $.99, respectively. Do these prices fit your mathematical model? If not, are they higher than predicted or lower?<BR>\n" );
document.write( "d) Suppose that you found cartons of milk marked at $3.45, but that there was nothing on the carton to tell what size it is. According to your model, how much would such a carton hold?<BR>\n" );
document.write( "e) What doe the price-intercept represent in the real world?<BR>\n" );
document.write( "f) What are the units of the slope? What real-world quantity does this number represent?<BR>\n" );
document.write( "g) Sketch and label the graph. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #412182 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=412182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = number of quarts<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > = price in $<BR>\n" );
document.write( " <BR>\n" );
document.write( "(Explanation: We have to start by defining variables that will be related by a linear equation. Next, we have to use the two (x,y) data points given to find the equation)<BR>\n" );
document.write( " <BR>\n" );
document.write( "one gallon (4 quarts) for $3.09 --> (4,3.09) (x=4, y=3.09)<BR>\n" );
document.write( "half gallon (2 quarts) for $1.65 --> (2,1.65) (x=2, y=1.65)<BR>\n" );
document.write( " <BR>\n" );
document.write( "a) Write the particular equation expressing price in terms of quarts.<BR>\n" );
document.write( "(Explanation: We are looking for an equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We can substitute the (x,y) values for each data point to get a system of equation to solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Otherwise, we can use the two points to calculate the slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the line, and then we can find the rest of the equation one way or another)\r<BR>\n" );
document.write( "\n" );
document.write( "Using a system of equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283.09=4m%2Bb%2C1.65=2m%2Bb%29 ALIGN=MIDDLE  ALT=\"system%283.09=4m%2Bb%2C1.65=2m%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting the second equation from the first equation;<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.09=4m%2Bb ALIGN=MIDDLE  ALT=\"3.09=4m%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.65=2m%2Bb ALIGN=MIDDLE  ALT=\"1.65=2m%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.44=2m ALIGN=MIDDLE  ALT=\"1.44=2m\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.44%2F2=2m%2F2 ALIGN=MIDDLE  ALT=\"1.44%2F2=2m%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28m=0.72%29 ALIGN=MIDDLE  ALT=\"highlight%28m=0.72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting that value in the second equation, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.65=2%2A0.72%2Bb ALIGN=MIDDLE  ALT=\"1.65=2%2A0.72%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.65=1.44%2Bb ALIGN=MIDDLE  ALT=\"1.65=1.44%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.65-1.44=1.44%2Bb-1.44 ALIGN=MIDDLE  ALT=\"1.65-1.44=1.44%2Bb-1.44\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280.21=b%29 ALIGN=MIDDLE  ALT=\"highlight%280.21=b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=0.72x%2B0.21%29 ALIGN=MIDDLE  ALT=\"highlight%28y=0.72x%2B0.21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Using slope:<BR>\n" );
document.write( "(Explanation: the slope, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the change in y divided by the change in x, as we go from one data point to the other.)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%283.09-1.65%29%2F%284-2%29 ALIGN=MIDDLE  ALT=\"m=%283.09-1.65%29%2F%284-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1.44%2F2 ALIGN=MIDDLE  ALT=\"m=1.44%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28m=0.72%29 ALIGN=MIDDLE  ALT=\"highlight%28m=0.72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The point-slope form of the equation, using point (2,1.65), is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-1.65=0.72%28x-2%29 ALIGN=MIDDLE  ALT=\"y-1.65=0.72%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-1.65=0.72%28x-2%29 ALIGN=MIDDLE  ALT=\"y-1.65=0.72%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-1.65=0.72x-1.44%29 ALIGN=MIDDLE  ALT=\"y-1.65=0.72x-1.44%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-1.65%2B1.65=0.72x-1.44%2B1.65%29 ALIGN=MIDDLE  ALT=\"y-1.65%2B1.65=0.72x-1.44%2B1.65%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=0.72x%2B0.21%29 ALIGN=MIDDLE  ALT=\"highlight%28y=0.72x%2B0.21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "b) If Handy Andy sold 3-gallon cartons, what would your equation predict the price to be?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3gallons=3%2A4quarts=12quarts ALIGN=MIDDLE  ALT=\"3gallons=3%2A4quarts=12quarts\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=12 ALIGN=MIDDLE  ALT=\"x=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the equation, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.72%2A12%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.72%2A12%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=8.64%2B0.21 ALIGN=MIDDLE  ALT=\"y=8.64%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=8.85%29 ALIGN=MIDDLE  ALT=\"highlight%28y=8.85%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Andy would sell a 3-gallon carton for $8.85. <BR>\n" );
document.write( " <BR>\n" );
document.write( "c) The actual prices for pint cartons (1/2 quart) and one-quart cartons are $.57 and $.99, respectively. Do these prices fit your mathematical model? If not, are they higher than predicted or lower?<BR>\n" );
document.write( "The predictions are found using the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.72x%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.72x%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For 1/2 quart = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.5 ALIGN=MIDDLE  ALT=\"0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > quart, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0.5 ALIGN=MIDDLE  ALT=\"x=0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.72%2A0.5%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.72%2A0.5%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.36%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.36%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=0.57%29 ALIGN=MIDDLE  ALT=\"highlight%28y=0.57%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For 1 quart, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.72%2A1%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.72%2A1%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.72%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.72%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=0.93%29 ALIGN=MIDDLE  ALT=\"highlight%28y=0.93%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the price for pint cartons agrees with the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.57 ALIGN=MIDDLE  ALT=\"y=0.57\"   DISABLED_style= \"max-width:90%; height:auto;\" > prediction,<BR>\n" );
document.write( "but the price for the quart, $0.99, is higher than the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.93 ALIGN=MIDDLE  ALT=\"y=0.93\"   DISABLED_style= \"max-width:90%; height:auto;\" > prediction.<BR>\n" );
document.write( " <BR>\n" );
document.write( "d) Suppose that you found cartons of milk marked at $3.45, but that there was nothing on the carton to tell what size it is. According to your model, how much would such a carton hold?<BR>\n" );
document.write( "I would use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3.45 ALIGN=MIDDLE  ALT=\"y=3.45\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.72x%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.72x%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the number of quarts, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.45=0.72x%2B0.21 ALIGN=MIDDLE  ALT=\"3.45=0.72x%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.45-0.21=0.72x%2B0.21-0.21 ALIGN=MIDDLE  ALT=\"3.45-0.21=0.72x%2B0.21-0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.24=0.72x ALIGN=MIDDLE  ALT=\"3.24=0.72x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.24%2F0.72=x ALIGN=MIDDLE  ALT=\"3.24%2F0.72=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=4.5%29 ALIGN=MIDDLE  ALT=\"highlight%28x=4.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Such a carton would hold 4.5 quarts according to the model.<BR>\n" );
document.write( " <BR>\n" );
document.write( "e) What does the price-intercept represent in the real world?<BR>\n" );
document.write( "The price-intercept is the y-intercept in the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.72x%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.72x%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "It is the value, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.21 ALIGN=MIDDLE  ALT=\"0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" >, representing $0.21, the price of a carton with zero quarts of milk inside. It represents the cost of the carton plus maybe Andy's work making a sale.<BR>\n" );
document.write( " <BR>\n" );
document.write( "f) What are the units of the slope? What real-world quantity does this number represent?<BR>\n" );
document.write( "The slope, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.72 ALIGN=MIDDLE  ALT=\"0.72\"   DISABLED_style= \"max-width:90%; height:auto;\" >, represents the cost of the milk per quart.<BR>\n" );
document.write( "The units are $/quart, and it gets multiplied by the number of quarts to find the price of the milk inside the carton in $.\r<BR>\n" );
document.write( "\n" );
document.write( "g) Sketch and label the graph.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C14%2C-1%2C9%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28line%280%2C0.21%2C14%2C10.29%29%29%2C%0D%0Ablue%28circle%280%2C0.21%2C0.2%29%29%2Cblue%28circle%284%2C3.09%2C0.2%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > To graph, we just need to plot 2 points (blue circles) and connect then with a line.<BR>\n" );
document.write( "The blue line represents <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.72x%2B0.21 ALIGN=MIDDLE  ALT=\"y=0.72x%2B0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The blue circles represent<BR>\n" );
document.write( "the intercept point, (0,0.21), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0.21 ALIGN=MIDDLE  ALT=\"y=0.21\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and<BR>\n" );
document.write( "point (4,3.09), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3.09 ALIGN=MIDDLE  ALT=\"y=3.09\"   DISABLED_style= \"max-width:90%; height:auto;\" >, representing 4 quarts at $3.09. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );