]]> As described last week, Lester has a big red barn. He just got a llama named Latisha. ]]> But Latisha needs a pen so she doesn't wander the neighborhood. Lester doesn't think the neighbors would appreciate the little presents she might leave in their lawns. ]]> So, Lester buys 36 meters of fencing from Orchard Depot. Because of your renowned mathematical skills, he hires you to design the perfect pen with the fencing. ]]> {1} Point(3,3)[hidden]; {2} Point(247,3)[hidden]; {3} Point(29,159)[hidden]; {4} Point(200,159)[hidden]; {5} Point(413,141)[hidden]; {6} Segment(2,1)[hidden,black]; {7} Segment(4,3)[hidden,thick,red]; {8} Distance(4,3,354,254,'available fence ')[blue]; {9} UnitPoint(5,18.34646606)[hidden]; {10} Perpendicular(6,1)[hidden,black]; {11} Point on object(6,0.19755585)[hidden]; {12} Origin&Unit(5,9)[hidden,blue]; {13} Perpendicular(6,2)[hidden,black]; {14} AxisX(12)[hidden,blue]; {15} AxisY(12)[hidden,blue]; {16} Point on object(10,0.69509107)[hidden,thick]; {17} Parallel(6,16)[hidden,thick,red]; {18} Intersect(17,13)[hidden,thick]; {19} Polygon(2,18,16,1)[thick,red]; {1} Point(413,141)[hidden]; {2} Point(200,159)[hidden]; {3} Point(29,159)[hidden]; {4} Point(202,54)[hidden]; {5} Point(-42,54)[hidden]; {6} UnitPoint(1,18.34646606)[hidden]; {7} Distance(2,3,354,254,'available fence ')[blue]; {8} Segment(2,3)[hidden,thick,red]; {9} Segment(4,5)[hidden,black]; {10} Perpendicular(9,4)[hidden,black]; {11} Origin&Unit(1,6)[hidden,blue]; {12} Point on object(9,0.19755585)[hidden]; {13} Point on object(8,0.29371941)[hidden]; {14} Perpendicular(9,5)[hidden,black]; {15} Point on object(14,0.69509107)[hidden,thick]; {16} AxisY(11)[hidden,blue]; {17} AxisX(11)[hidden,blue]; {18} Segment(3,13)[hidden,thick,blue]; {19} Parallel(9,15)[hidden,thick,red]; {20} Circle by radius(13,18)[hidden,red]; {21} Circle by radius(2,18)[hidden,red]; {22} Length(18,354,267,'Length(Segment height) = ')[blue]; {23} Intersect(19,10)[hidden,thick]; {24} Intersect1(8,21)[hidden]; {25} Translation/FixedAngle/MarkedDistance(12,22,1.57079637)[hidden]; {26} Segment(2,24)[hidden,thick,blue]; {27} Segment(24,13)[hidden,thick,yellow]; {28} Segment(25,12)[thick,blue]; {29} Length(27,354,280,'Length(Segment width) = ')[blue]; {30} Translation/FixedAngle/MarkedDistance(25,29,0.00000000)[hidden]; {31} Segment(30,25)[thick,yellow]; {32} Perpendicular(9,30)[hidden,blue]; {33} Intersect(32,9)[hidden]; {34} Segment(33,30)[thick,blue]; {35} Polygon(12,25,30,33)[green]; {1} Point(177,10)[hidden]; {2} Point(6,10)[hidden]; {3} Segment(1,2)[hidden,black]; {4} Midpoint(3)[hidden,thick]; {5} Segment(1,4)[hidden,thick,red]; {6} Point on object(5,0.41520467)[hidden,thick]; {7} Circle(4,6)[hidden,thick,red]; {8} Intersect1(3,7)[hidden,thick]; {9} Intersect2(3,7)[hidden,thick]; {10} Segment(2,8)[thick,blue]; {11} Segment(6,8)[thick,yellow]; {12} Segment(1,9)[thick,blue]; {13} Segment(9,4)[hidden,black]; {1} Point(413,141)[hidden]; {2} Point(200,159)[hidden]; {3} Point(29,159)[hidden]; {4} Point(250,75)[hidden]; {5} Point(6,75)[hidden]; {6} UnitPoint(1,18.34646606)[hidden]; {7} Distance(2,3,354,254,'available fence ')[blue]; {8} Segment(2,3)[hidden,thick,red]; {9} Segment(4,5)[hidden,black]; {10} Perpendicular(9,4)[hidden,black]; {11} Origin&Unit(1,6)[hidden,blue]; {12} Point on object(9,0.19755585)[hidden]; {13} Point on object(8,0.41652644)[hidden]; {14} Perpendicular(9,5)[hidden,black]; {15} Point on object(14,0.69509107)[hidden,thick]; {16} AxisY(11)[hidden,blue]; {17} AxisX(11)[hidden,blue]; {18} Segment(3,13)[hidden,thick,blue]; {19} Parallel(9,15)[hidden,thick,red]; {20} Circle by radius(13,18)[hidden,red]; {21} Circle by radius(2,18)[hidden,red]; {22} Length(18,354,267,'Length(Segment height) = ')[blue]; {23} Intersect(19,10)[hidden,thick]; {24} Intersect1(8,21)[hidden]; {25} Translation/FixedAngle/MarkedDistance(12,22,1.57079637)[hidden]; {26} Polygon(4,23,15,5)[thick,red]; {27} Segment(2,24)[hidden,thick,blue]; {28} Segment(24,13)[hidden,thick,yellow]; {29} Segment(25,12)[thick,blue]; {30} Length(28,354,280,'Length(Segment width) = ')[blue]; {31} Translation/FixedAngle/MarkedDistance(25,30,0.00000000)[hidden]; {32} Segment(31,25)[thick,yellow]; {33} Perpendicular(9,31)[hidden,blue]; {34} Intersect(33,9)[hidden]; {35} Segment(34,31)[thick,blue]; Lester wants you to use the fencing for 3 sides of the pen, and use the long side of his barn as the fourth side. He wants the pen in the shape of a rectangle, but he needs you to figure out just how wide or narrow it should be. ]]> Now Latisha the llama isn't picky, but she is ravenous! She needs the largest possible area in which to graze.
Go to the next page to help Lester design the pen with the largest possible area. ]]>

]]> What pattern do you notice when you plot width vs. area?
Use the red dot to change the amount of fencing used for height versus width. Click the Sample button to plot the current width vs. area. Click Clear to erase the whole graph.
Experiment with many different widths to find the largest area and then answer the questions in the notepad below. ]]> {1} Point(207,144)[hidden]; {2} Point(36,144)[hidden]; {3} Point(249,87)[hidden]; {4} Point(5,87)[hidden]; {5} Distance(1,2,268,197,'available fence ')[blue]; {6} Segment(1,2)[hidden,black]; {7} Segment(3,4)[hidden,black]; {8} Perpendicular(7,3)[hidden,black]; {9} Perpendicular(7,4)[hidden,black]; {10} Midpoint(6)[hidden,thick]; {11} Point on object(9,0.69509107)[hidden,thick]; {12} Segment(1,10)[hidden,thick,red]; {13} Parallel(7,11)[hidden,black]; {14} Point on object(12,0.28654972)[thick]; {15} Intersect(13,8)[hidden,thick]; {16} Circle(10,14)[hidden,thick,red]; {17} Polygon(3,15,11,4)[thick,red]; {18} Intersect2(6,16)[thick]; {19} Intersect1(6,16)[hidden,thick]; {20} Perpendicular(6,18)[hidden,thick,red]; {21} Segment(18,10)[hidden,black]; {22} Segment(1,18)[thick,blue]; {23} Perpendicular(6,19)[hidden,black]; {24} Segment(14,19)[thick,yellow]; {25} Segment(2,19)[thick,blue]; {26} Intersect(7,20)[hidden]; {27} Intersect(7,23)[hidden]; {28} Length(24,268,223,'Length(Segment width) = ')[black]; {29} Length(25,268,210,'Length(Segment height) = ')[black]; {30} Circle by radius(27,25)[hidden,black]; {31} Intersect1(23,30)[hidden]; {32} Intersect2(23,30)[hidden]; {33} Segment(31,27)[thick,blue]; {34} Parallel(7,31)[hidden,black]; {35} Intersect(20,34)[hidden]; {36} Segment(35,31)[thick,yellow]; {37} Segment(26,35)[thick,blue]; {38} Point on object(36,0.69444448)[hidden,thick]; {39} Point on object(37,0.33333337)[hidden,thick]; {40} AnimateButton(267,162,'Animate')(39,37,38,36)(1,1)(0,0)(0,0)[black,auto]; {41} Perpendicular(36,38)[hidden,black]; {42} Perpendicular(37,39)[hidden,black]; {43} Intersect(41,42); 11.368421052631579 13.26315789473684 150.78116343490302 54.0 63.0 171.0 ((0,0,0)(13.26315789473684,150.78116343490302),(8.210526315789473,114.08310249307478),(5.684210526315789,86.1606648199446),(2.7368421052631575,45.5180055401662),(30.105263157894736,88.73130193905818),(25.894736842105264,130.8365650969529),(20.42105263157895,159.0692520775623),(15.789473684210526,159.5567867036011))