document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=822055>Question 822055</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are to design a fountain for the garden of one of Emily’s customers.<BR>\n" );
document.write( "• The fountain will have a cylindrical base with a cone on top.<BR>\n" );
document.write( "• The cylindrical base will have a diameter of 1 m.<BR>\n" );
document.write( "• The fountain is to be made of concrete.<BR>\n" );
document.write( "• The entire fountain is to be coated with protective paint.<BR>\n" );
document.write( "a) Make a sketch of your design, showing all<BR>\n" );
document.write( "dimensions.<BR>\n" );
document.write( "b) How much concrete is needed to make the<BR>\n" );
document.write( "fountain?<BR>\n" );
document.write( "c) What is the surface area that needs to be<BR>\n" );
document.write( "painted?<BR>\n" );
document.write( "d) Concrete costs $100/m3. Each litre of<BR>\n" );
document.write( "protective paint costs $17.50 and covers <BR>\n" );
document.write( "5 m2. Find the total cost of the materials<BR>\n" );
document.write( "needed to make the fountain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #494605 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=494605\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Not enough data, and not reasonable data.<BR>\n" );
document.write( "You must be missing some details of the problem.<BR>\n" );
document.write( "A 1 meter wide cylindrical base means a monstrously large fountain. It would be appropriate for a city square, but not for the garden of any citizen.<BR>\n" );
document.write( "A 1 meter diameter would be appropriate for the top cone of the fountain in someone's garden,<BR>\n" );
document.write( "but even assuming 1 meter diameter and 1 meter base height, there are other design issues that are not specified.<BR>\n" );
document.write( "Is the top cone part a solid cone?<BR>\n" );
document.write( "If there is a cavity in the cone, is the wall of the cone uniformly thick? How thick?<BR>\n" );
document.write( "How wide is the base?<BR>\n" );
document.write( "Is the base a hollow stem? Or is it a solid cylinder made of concrete?<BR>\n" );
document.write( "If it is hollow, how thick is the wall of the base cylinder?<BR>\n" );
document.write( "Depending on design, it could be a complicated problem.<BR>\n" );
document.write( "There is also the issue of how realistic you have to be.<BR>\n" );
document.write( "Do you need to buy a whole number of liters of paint, or can you buy exactly the amount you need, even if it is a crazy decimal number, such as 0.17 liter?<BR>\n" );
document.write( "What about the cement? Can you order 0.43 m^3?<BR>\n" );
document.write( " <BR>\n" );
document.write( "The simplest, if ugly, design would be a cylinder with a conical hollow at the top.<BR>\n" );
document.write( "In cross section, it would look like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-55%2C55%2C-10%2C110%2C%0D%0Aline%28-50%2C0%2C50%2C0%29%2Cline%28-50%2C0%2C-50%2C100%29%2Cline%2850%2C0%2C50%2C100%29%2C%0D%0Aline%28-50%2C100%2C0%2C75%29%2Cline%2850%2C100%2C0%2C75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "VOLUME:<BR>\n" );
document.write( "The volume of that is the volume of the cylinder minus the volume of the cone.<BR>\n" );
document.write( "If the cylinder and cone were 1 m in diameter (0.5m radius), with a cylinder height of 1m, and a cone height (depth) of 0.25m, the volume, in m^3 would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A0.5%5E2%2A1-pi%2A0.5%5E2%2A1%2F3=0.25pi-0.25pi%2F3=0.5236 ALIGN=MIDDLE  ALT=\"pi%2A0.5%5E2%2A1-pi%2A0.5%5E2%2A1%2F3=0.25pi-0.25pi%2F3=0.5236\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At $100/m^3, the cement to make that would cost $52.36.<BR>\n" );
document.write( " <BR>\n" );
document.write( "SURFACE AREAS:<BR>\n" );
document.write( "The surface to be painted would be the lateral surface of the cylinder,<BR>\n" );
document.write( "calculates as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter%2Aheight=pi%2Adiameter%2Aheight ALIGN=MIDDLE  ALT=\"perimeter%2Aheight=pi%2Adiameter%2Aheight\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "plus the lateral surface of the cone,<BR>\n" );
document.write( "calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter%2Aheight%2F2=pi%2Adiameter%2Aheight ALIGN=MIDDLE  ALT=\"perimeter%2Aheight%2F2=pi%2Adiameter%2Aheight\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the measurements assumed above, the area in m^2 would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A1%2A1%2Bpi%2A1%2A0.25%2F2=pi%2Bpi%2F3=4pi%2F3=4.19 ALIGN=MIDDLE  ALT=\"pi%2A1%2A1%2Bpi%2A1%2A0.25%2F2=pi%2Bpi%2F3=4pi%2F3=4.19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "COST:<BR>\n" );
document.write( "The cost for 1 liter of paint and 0.53m^3 of concrete would be<BR>\n" );
document.write( "$17.50 + $53.00 = $70.50<BR>\n" );
document.write( " <BR>\n" );
document.write( "The fountain I would design, in cross-section, would look like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C400%2C-60%2C60%2C-20%2C140%2C%0D%0Arectangle%28-10%2C0%2C-8%2C100%29%2Crectangle%2810%2C0%2C8%2C100%29%2C%0D%0Aline%280%2C96%2C-50%2C121%29%2Cline%280%2C96%2C50%2C121%29%2C%0D%0Aline%28-50%2C121%2C-45%2C121%29%2Cline%2850%2C121%2C45%2C121%29%2C%0D%0Aline%280%2C98.5%2C-45%2C121%29%2Cline%280%2C98.5%2C45%2C121%29%2C%0D%0Aarrow%28-5%2C127%2C-50%2C127%29%2Carrow%285%2C127%2C50%2C127%29%2C%0D%0Alocate%28-4%2C130%2C1m%29%2Clocate%2811%2C53%2C1m%29%2C%0D%0Aarrow%2814%2C54%2C14%2C100%29%2Carrow%2814%2C46%2C14%2C0%29%2C%0D%0Aarrow%280%2C-3%2C-10%2C-3%29%2Carrow%280%2C-3%2C10%2C-3%29%2C%0D%0Alocate%28-8%2C-4%2C0.20m%29%2Clocate%28-8%2C19%2C.16cm%29%2C%0D%0Aarrow%280%2C20%2C-8%2C20%29%2Carrow%280%2C20%2C8%2C20%29%2C%0D%0Aarrow%28-10%2C121%2C-44.5%2C121%29%2Carrow%2810%2C121%2C44.5%2C121%29%2C%0D%0Alocate%28-10%2C124%2C0.90m%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The height of the cone is half the diameter.<BR>\n" );
document.write( " <BR>\n" );
document.write( "VOLUMES:<BR>\n" );
document.write( "The volume of cement in the cone is the volume of a cone with a 0.5m radius and a 0.25m height, minus the volume of a cone with a 0.45m radius and a 0.225m height. In <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E3 ALIGN=MIDDLE  ALT=\"m%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , it is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28pi%2A0.5%5E2%2A0.25-pi%2A0.45%5E2%2A0.225%29%2F3=pi%2A%280.06250-0.04556%29%2F3=pi%2A.01694%2F3=0.01774 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The volume of cement in the base is the volume of a cylinder with a 0.1m radius and a 1m height, minus the volume of a cone with a 0.08m radius and a 1m height. In <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E3 ALIGN=MIDDLE  ALT=\"m%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , it is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A0.1%5E2%2A1-pi%2A0.08%5E2%2A1=pi%28.01-.0064%29=0.036pi=0.01131 ALIGN=MIDDLE  ALT=\"pi%2A0.1%5E2%2A1-pi%2A0.08%5E2%2A1=pi%28.01-.0064%29=0.036pi=0.01131\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The total volume of concrete is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.01774m%5E3%2B0.01131m%5E3=0.02905m%5E3 ALIGN=MIDDLE  ALT=\"0.01774m%5E3%2B0.01131m%5E3=0.02905m%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "At $100/m^3 that would cost $2.91.<BR>\n" );
document.write( " <BR>\n" );
document.write( "SURFACE AREAS:<BR>\n" );
document.write( "The outside cone has a radius of 0.5m and a height of 0.25m.<BR>\n" );
document.write( "Its slant height, in m, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%280.5%5E2%2B0.25%5E2%29=sqrt%280.25%2B0.0625%29=sqrt%280.3125%29=0.559 ALIGN=MIDDLE  ALT=\"sqrt%280.5%5E2%2B0.25%5E2%29=sqrt%280.25%2B0.0625%29=sqrt%280.3125%29=0.559\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perimeter of the base, in m, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because the diameter is 1m.<BR>\n" );
document.write( "The outside surface area of the cone, in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2 ALIGN=MIDDLE  ALT=\"m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A0.559%2F2=0.878 ALIGN=MIDDLE  ALT=\"pi%2A0.559%2F2=0.878\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The surface area of the cone insides plus the rim is about the same.<BR>\n" );
document.write( "The outside surface area of the cylindrical base, in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2 ALIGN=MIDDLE  ALT=\"m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A0.2%2A1=0.628 ALIGN=MIDDLE  ALT=\"pi%2A0.2%2A1=0.628\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The total area to be painted, in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2 ALIGN=MIDDLE  ALT=\"m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is about<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.878%2B0.878%2B0.628=2.38 ALIGN=MIDDLE  ALT=\"0.878%2B0.878%2B0.628=2.38\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That would require less than 1 liter of paint.<BR>\n" );
document.write( " <BR>\n" );
document.write( "COST:<BR>\n" );
document.write( "Buying 1 liter of paint, plus 0.03m^3 concrete would cost<BR>\n" );
document.write( "$17.50 + $3.00 = $20.50 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );