document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=560284>Question 560284</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A common ant absorbs oxygen at a rate of about 6.2 mililiters per second per square centimeter of exoskeleton. It needs about 24 mililiters of oxygen per second per cubic centimeter of its body. An ant is basically cylindrical in shape, so its surface area S and volume V can be approximated by the formulas for the surface area and volume of a cylinder:\r<BR>\n" );
document.write( "\n" );
document.write( "S=2&#960;h+2&#960;r^2<BR>\n" );
document.write( "V=&#960;r^2h\r<BR>\n" );
document.write( "\n" );
document.write( "a) Approximate the surface area and volume of an ant that is 8 mm long and has a radius of 1.5 mm. would this ant have a surface area large enough to meet its oxygen needs?<BR>\n" );
document.write( "b) Consider a \"giant\" ant that is 8m long and has a radius of 1.5m. would this ant have a surface area large enough to meet its oxygen needs?<BR>\n" );
document.write( "c) substitute 1000r for r and 1000h for h into the formulas for surface area and volume. how does increasing the radius by height by a factor of 1000 effect surface area> How does it affect volume? Use the results to explain why \"giant\" ants do not exist. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #363815 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=363815\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The formula to calculate surface area of a cylinder is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=2pi%2Arh%2B2pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"S=2pi%2Arh%2B2pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (You had a typo).<BR>\n" );
document.write( "The formula to calculate its volume is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%2Ar%5E2h ALIGN=MIDDLE  ALT=\"V=pi%2Ar%5E2h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the measurements in cm, square centimeters (sq cm), and cubic centimeters (cc) would be best.<BR>\n" );
document.write( "1 mm = 0.1 cm, so 8 mm = 0.8 cm and 1.5 mm = 0.15 cm (you just divide by 10)<BR>\n" );
document.write( "1 m = 100 cm, so 8 m = 800 cm and 1.5 m = 150 cm (you just multiply times 100)<BR>\n" );
document.write( "Your teacher probably expects you to use the approximation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >=3.14, so I'll use that.<BR>\n" );
document.write( "a)  For a cylindrical ant that is 8 mm long and has a radius of 1.5 mm:<BR>\n" );
document.write( "S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2A0.15%2A0.8%2B2pi%2A0.15%5E2 ALIGN=MIDDLE  ALT=\"2pi%2A0.15%2A0.8%2B2pi%2A0.15%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = approx. 0.89 sq cm<BR>\n" );
document.write( "V = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A0.15%5E2%2A0.8 ALIGN=MIDDLE  ALT=\"pi%2A0.15%5E2%2A0.8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E3 ALIGN=MIDDLE  ALT=\"cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > = approx. 0.057 cc<BR>\n" );
document.write( "In one second, it would absorb <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.2%2A0.89%2A1 ALIGN=MIDDLE  ALT=\"6.2%2A0.89%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > mL oxygen = 5.5 mL oxygen.<BR>\n" );
document.write( "During that second it would require <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2A0.057%2A1 ALIGN=MIDDLE  ALT=\"24%2A0.057%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > mL oxygen = 1.37 mL oxygen.<BR>\n" );
document.write( "This ant would have a surface area large enough to meet its oxygen needs.<BR>\n" );
document.write( "b) For a \"giant\" ant that is 8m long and has a radius of 1.5m (1,000 times longer in every direction):<BR>\n" );
document.write( "S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2A150%2A800%2B2pi%2A150%5E2 ALIGN=MIDDLE  ALT=\"2pi%2A150%2A800%2B2pi%2A150%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 890,000 sq cm = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8.9%2A10%5E5 ALIGN=MIDDLE  ALT=\"8.9%2A10%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > sq cm<BR>\n" );
document.write( "V = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A150%5E2%2A800 ALIGN=MIDDLE  ALT=\"pi%2A150%5E2%2A800\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E3 ALIGN=MIDDLE  ALT=\"cm%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > = approx. 57,000,000 cc = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5.7%2A10%5E7 ALIGN=MIDDLE  ALT=\"5.7%2A10%5E7\"   DISABLED_style= \"max-width:90%; height:auto;\" > cc<BR>\n" );
document.write( "In one second, it would absorb <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.2%2A8.9%2A10%5E5%2A1 ALIGN=MIDDLE  ALT=\"6.2%2A8.9%2A10%5E5%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > mL oxygen = 5,500,000 mL oxygen = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5.5%2A10%5E6 ALIGN=MIDDLE  ALT=\"5.5%2A10%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > mL oxygen.<BR>\n" );
document.write( "During that second it would require <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2A5.7%2A10%5E7%2A1 ALIGN=MIDDLE  ALT=\"24%2A5.7%2A10%5E7%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > mL oxygen = 1,370,000,000 mL oxygen = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.37%2A10%5E9 ALIGN=MIDDLE  ALT=\"1.37%2A10%5E9\"   DISABLED_style= \"max-width:90%; height:auto;\" > mL oxygen.<BR>\n" );
document.write( "This ant would not have a surface area large enough to meet its oxygen needs.<BR>\n" );
document.write( "c) Substituting 1000r for r and 1000h for h into the formulas for surface area and volume, we see that<BR>\n" );
document.write( "S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2A%281000r%29%281000h%29%2B2pi%2A%281000r%29%5E2 ALIGN=MIDDLE  ALT=\"2pi%2A%281000r%29%281000h%29%2B2pi%2A%281000r%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2C000%2C000%282pi%2Arh%2B2pi%2Ar%5E2%29 ALIGN=MIDDLE  ALT=\"1%2C000%2C000%282pi%2Arh%2B2pi%2Ar%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "V = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A%281000r%29%5E2%281000h%29 ALIGN=MIDDLE  ALT=\"pi%2A%281000r%29%5E2%281000h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2C000%2C000%2C000%28pi%2Ar%5E2h%29 ALIGN=MIDDLE  ALT=\"1%2C000%2C000%2C000%28pi%2Ar%5E2h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "With the same shape, as the length measurements increase by a factor of 1,000, the surface are increases by a factor of 1,000,000 (1,000^2), and the volume increases by a factor of 1,000,000,000 (1,000^3).<BR>\n" );
document.write( "NOTE: The same is true for any shape (not just cylinders), and any factor (not just 1,000). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );