探究无盖长方体的最大容积

第一张纸长宽的比值为1:1，也就是正方 形，为了保险起见，试两组数据，一组边长为 50cm，另一组边长为100cm 。用 excel 来做。正方形边长为 50cm宽思维作业在七年级上册的数学书中，有一个课题叫 做“探究无盖长方体的最大容积”，大致内容是： 一张纸如何剪能让它折起来之后的无盖长方体 的体积（也就是容积）最大。学习过后，思索 很久，得出以下结论。如何剪一张纸才能让它折起来之后体积最 大，说明白了也就是怎么样在这张纸的四个角 处剪相同的正方形才能让其体积最大，在其四 个角之处剪的正方形的边长也就是折起来之后 的长方体的高。如果设减去的小长方形的边长为 x（即长方体的高为X），那张纸的长为a，宽为b， 那么长方体的长就是（a2x），宽就是（b2x）， 因此四个小正方形的边长不能大于等于这张纸 的长和宽。今天要研究的是减去的四个小正方形的边长 是否与那张纸的长宽有关。（所有数值精确到0.1,因为有些四个小正 方形的边长可能会是循环小数，除下来的比值 也可能除不尽，故四舍五入到0.1）0.149.849.8248.0040.249.649.6492.0320.349.449.4732.1080.449.249.2968.2560.549491200.50.648.848.81428.8640.748.648.61653.3720.848.448.41874.0480.948.248.22090.9167.435.235.29168.8967.535359187.57.634.834.89203.9047.734.634.69218.132长体积7.834.434.49230.2087.934.234.29240.1568343492488.133.833.89253.7648.233.633.69257.4728.333.433.49259.1488.433.233.29258.8168.533339256.58.632.832.89252.2248.732.632.69246.0128.832.432.49237.8888.932.232.29227.8769323292169.131.831.89202.2849.231.631.69186.7529.331.431.49169.4289.431.231.29150.33622.94.24.2403.956234436823.13.83.8333.56423.23.63.6300.67223.33.43.4269.34823.43.23.2239.61623.533211.523.62.82.8185.02423.72.62.6160.21223.82.42.4137.08823.92.22.2115.676上表省略了一大部分，标红字处是它体积的最大值。8.3cm 约是在其四个角之处剪的正方形的边长的最大 值（使折成长方体后体积最大）。8.350=0.166=16.6%正方形边长为 100cm0.199.899.8996.0040.299.699.61984.0320.399.499.42964.1080.499.299.23936.2560.599994900.50.698.898.85856.8640.798.698.66805.3720.898.498.47746.0480.998.298.28678.91615.968.268.273954.721668687398416.167.867.874009.1216.267.667.674030.1116.367.467.474046.9916.467.267.274059.7816.5676774068.516.666.866.874073.1816.766.666.674073.8516.866.466.474070.5316.966.266.274063.241766667405217.165.865.874036.8417.265.665.674017.7917.365.465.473994.8717.465.265.273968.117.5656573937.517.664.864.873903.117.764.664.673864.9317.864.464.473823.0117.964.264.273777.3648.92.22.2236.676492219649.11.81.8159.08449.21.61.6125.95249.31.41.496.62849.41.21.271.13649.51149.549.60.80.831.74449.70.60.617.89249.80.40.47.96849.90.20.21.996宽长体积16.7cm 约是在其四个角之处剪的正方形的边长的最大值（使折成长方体后 体积最大）。16. 7F 100=0.167=16.7%第二张纸长宽的比值为 2:1 ，为了保险起见，试两组数据，一组长为 200cm 宽为100cm，另一组长为100cm，宽为50cm。长为200cm，宽为100cm。高长宽体积0.1199.899.81994.0040.2199.699.63976.0320.3199.499.45946.1080.4199.299.27904.2560.5199999850.50.6198.898.811784.860.7198.698.613707.370.8198.498.415618.050.9198.298.217516.9220.6158.858.8192351.320.7158.658.619238520.8158.458.4192411.620.9158.258.2192431.3211585819244421.1157.857.8192449.721.2157.657.6192448.521.3157.457.4192440.421.4157.257.2192425.421.515757192403.521.6156.856.8192374.848.9102.22.210994.68491022999649.1101.81.88997.08449.2101.61.67997.95249.3101.41.46998.62849.4101.21.25999.13649.510114999.549.6100.80.83999.74449.7100.60.62999.89249.8100.40.41999.96849.9100.20.2999.99621.1cm 约是在其四个角之 处剪的正方 形的边长的最大 值21.1100=0.211=21.1%长为100cm，宽为50cm。(10.55%+21.1%)F2=0.158250.16=16%0.199.849.8497.0040.299.649.6988.0320.399.449.41473.1080.499.249.21952.2560.599492425.50.698.848.82892.8640.798.648.63354.3720.898.448.43810.0480.998.248.24259.91610.179.829.824018.210.279.629.624032.8310.379.429.424043.9110.479.229.224051.4610.5792924055.510.678.828.824056.0610.778.628.624053.1710.878.428.424046.8510.978.228.224037.121178282402411.177.827.824007.5224522249624.151.81.82247.08424.251.61.61997.95224.351.41.41748.62824.451.21.21499.13624.55111249.524.650.80.8999.74424.750.60.6749.892宽长体积24.850.424.950.20.4 499.9680.2 249.996使折成长方体后体积最大）。21.1200=0.1055=10.55%10.6cm 约是在其四个角之处剪的正方形的边长的最大值（使折成长方体后体积 最大）。10.6100=0.106=10.6% 10.650=0.212=21.2%(10.6%+21.2%)F2=0.1590.16=16%第三张纸长宽的比值为 3:1，为了保险起见，试两组数据，一组长为 300cm 宽为100cm，另一组长为120cm，宽为30cm。长为300cm，宽为100cm。高 长 宽 体积0.1299.899.82992.0040.2299.699.65968.0320.3299.499.48928.1080.4299.299.211872.260.52999914800.50.6298.898.817712.860.7298.698.620609.370.8298.498.423490.050.9298.298.226354.9222.1255.855.8315447.422.2255.655.6315492.222.3255.455.4315526.322.4255.255.2315549.722.525555315562.522.6254.854.8315564.722.7254.654.6315556.322.8254.454.4315537.422.9254.254.2315508232545431546823.1253.853.8315417.648.9202.22.221752.684920221979649.1201.81.817835.0849.2201.61.615869.9549.3201.41.413900.6349.4201.21.211927.1449.520119949.549.6200.80.87967.74449.7200.60.65981.89249.8200.40.43991.96849.9200.20.21997.99622.6cm 约是在其四个角之处剪的正方形的边长的最大值(使折成长方体后 体积最大)。22.6 F 3000.075=7.5%22.6 F 100=0.226=22.6%(7.5%+22.6%)F2=0.150515%长为120cm，宽为30cm。长宽体积0.1119.829.8357.0040.2119.629.6708.0320.3119.429.41053.1080.4119.229.21392.2560.5119291725.50.6118.828.82052.8640.7118.628.62374.3720.8118.428.42690.0480.9118.228.22999.9166.4107.217.211800.586.51071711823.56.6106.816.811841.986.7106.616.611856.056.8106.416.411865.736.9106.216.211871.04710616118727.1105.815.811868.647.2105.615.611860.997.3105.415.411849.077.4105.215.211832.914922257614.191.81.82329.88414.291.61.62081.15214.391.41.41829.82814.491.21.21575.93614.59111319.514.690.80.81060.54414.790.60.6799.09214.890.40.4535.16814.990.20.2268.7967cm 约是在其四个角之处剪的正方形的边长的最大值(使折成长方体后体 积最大)。7 F 1200.058=5.8%7F300.233=23.3%(5.8%+23.3%)F20.15=15%仅仅用 1:1, 2:1, 3:1 来举例，通过以上的计算，用在其四个角之处剪的 正方形的边长除以这个长方形的长、宽，最后把除得的商的平均数求出【如果设 减去的小长方形的边长为x(即长方体的高为x),那张纸的长为a,宽为b,则就 是 (x/a+x/b)】，发现平均数在15%17%之间浮动，纸的比例如何变都不影响。所以我们之后就可以用这一规律来剪一张纸才能让它折起来之后体积最大，算法是：0.5 (x/a+x/b) =16%x/a+x/b=032%ab(x/a+x/b)=32%abbx+ax=32%abx(a+b)= 32%abx=32%ab/(a+b)语言叙述：用长乘宽的积乘 32%再除以长宽的和即可求出其四个角之处剪的正方形的边长，按此数据剪，再折成长方体，这样它的体积就最大了。