季六级培优班 数学讲义.doc

2014年 春季六年级培优班 数学讲义第 2 讲 圆柱一、教学衔接二、教学内容一）【回顾旧知】二）【传授新课】1、 知识归纳圆柱1. 圆柱的形成：圆柱是以长方形的一边为轴旋转而得到的。 圆柱也可以由长方形卷曲而得到。（两种方式：1.以长方形的长为底面周长，宽为高;2.以长方形的宽为底面周长，长为高。其中，第一种方式得到的圆柱体体积较大。）2圆柱的高是两个底面之间的距离，一个圆柱有无数条高，他们的数值是相等的。3.圆柱的切割：a.横切：切面是圆，表面积增加2倍底面积，即S增=2R2b.竖切（过直径）：切面是长方形（如果h=2R，切面为正方形），该长方形的长是圆柱的高，宽是圆柱的底面直径，表面积增加两个长方形的面积，即S增=4Rh4. 圆柱的侧面展开图：a 沿着高展开,展开图形是长方形，如果h=2R，展开图形为正方形。 b. 不沿着高展开，展开图形是平行四边形或不规则图形。 C.无论如何展开都得不到梯形5：圆柱的相关计算公式：a.底面积：S底=R2 b. 底面周长：C=d=2R c. 侧面积：S侧=2Rh d表面积 ：S=2S底+S侧 =2R2+2Rh e体积：V=R2 h考试常见题型：a 已知圆柱的底面积和高，求圆柱的侧面积，表面积，体积，底面周长b已知圆柱的底面周长和高，求圆柱的侧面积，表面积，体积，底面积c已知圆柱的底面周长和体积，求圆柱的侧面积，表面积，高，底面积d已知圆柱的底面面积和高，求圆柱的侧面积，表面积，体积，e已知圆柱的侧面积和高，求圆柱的底面半径，表面积，体积，底面积以上几种常见题型的解题方法，通常是求出圆柱的底面半径和高，再根据圆柱的相关计算公式进行计算。2、 例题讲解例1、用一块长60厘米、宽40厘米的铁皮做圆柱形水桶的侧面，另找一块铁皮做底。这样做成的铁桶的容积最大是多少？（精确到1厘米3）分析与解：铁桶有以60厘米的边为高和以40厘米的边为高两种做法。以60厘米的边为高时，桶底周长为40厘米。由2r=40可知，r=。此时桶的容积是rh=（）×60=（厘米）以40厘米为高时，桶底周长为60厘米。由2r=60可知，r=。此时桶的容是rh=（）×60=（厘米）比较可知，容积最大是11465（厘米）例2、有一个圆柱体的零件，高10厘米，底面直径是6厘米，零件的一端有一个圆柱形的圆孔，圆孔的直径是4厘米，孔深5厘米（见下图）。如果将这个零件接触空气的部分涂上防锈漆，那么一共要涂多少平方厘米？ 分析与解：需要涂漆的面有圆柱体的下底面、外侧面、上面的圆环、圆孔的侧面、圆孔的底面，其中上面的圆环与圆孔的底面可以拼成一个与圆柱体的底面相同的圆。涂漆面积为×（）×2+×6×10+×4×5=×（18+60+20）=3.14×98=307.72三）【课堂练习】一、填空（1）圆柱的底面周长和高相等时，展开后的侧面一定是个（ ）。（2）一个圆柱体，两底面之间的距离是10厘米，底面周长是31.4厘米，把这个圆柱体的侧面展开得到一个长方形，长方形的周长是（ ）。（3）把一个圆柱的侧面展开，是一个边长9.42dm的正方形，这个圆柱的底面直径是（ ）。（4）圆柱两个底面的直径（ ）。把一个底面积为6.28平方厘米的圆柱，切成两个圆柱，表面积增加（ ）平方厘米。（5）把一根圆柱形木料据成四段，增加的底面有（ ）个。（6）一根圆柱形有机玻璃棒，体积是54立方厘米，底面积是4立方厘米，把它平均截成5段，每段长（ ）cm。二、解答题1、把两个底面直径都是4厘米，长都是4分米圆柱形钢材焊接成一个长的圆柱形钢材，焊接成的圆柱形钢材的表面积比原来两个小圆柱形钢材的表面积之和减少了多少？2、一个圆柱的侧面积是25.12平方厘米，底面半径是2厘米，它的表面积是多少？3、一个没有盖的圆柱形水桶，高5分米，底面周长是12.56分米，做2个这样的水桶大约要用多少铁皮？装桶的一担水有多重？（每立方分米水重1千克）4、一根圆柱形钢材长2米，如果把它锯成两段，表面积比原来增加6.28平方分米，求这根2米长钢材的质量。（每立方分米钢重7.8千克）5、用铁皮制作2个圆柱形水桶（无盖），底面半径为12厘米，高为35厘米。制作这样2个水桶需要用铁皮多少平方分米？这2个桶最多可盛水多少升？6、一根圆柱形钢材，截下1米。量的它的横截面的直径是20厘米，截下的体积占这根钢材的，这根钢材原来的体积是多少立方分米？三、课堂小结·回顾反思四、布置作业1、把一个高3分米的圆柱体底面平均分成若干个小扇形，然后把圆柱体切开，拼成一个与它等底等高的近似长方体，表面积比原来增加了120平方厘米，求圆柱体的体积。2、一根长2m的圆柱形木头，截去2分米的一段小圆柱后，表面积减少了12.56平方分米，那么这根木头原来的体积是多少？3、用一块长6.28厘米、宽3.14厘米的铁皮做圆柱形水桶的侧面，另找一块铁皮做底。这样做成的铁桶的容积最大是多少？4、将一块长10cm、宽6cm、高8cm的长方体木块，切割成体积尽可能大的圆柱体木块，求这个圆柱体木块的体积。.5、一个底面积是10平方厘米的圆柱，侧面展开后是一个正方形，求这个圆柱的侧面积。Winger Tuivasa-Sheck, who scored two tries in the Kiwis' 20-18 semi-final win over England, has been passed fit after a lower-leg injury, while Slater has been named at full-back but is still recovering from a knee injury aggravated against USA.Both sides boast 100% records heading into the encounter but Australia have not conceded a try since Josh Charnley's effort in their first pool match against England on the opening day.Aussie winger Jarryd Hayne is the competition's top try scorer with nine, closely followed by Tuivasa-Sheck with eight.But it is recently named Rugby League International Federation player of the year Sonny Bill Williams who has attracted the most interest in the tournament so far.The Kiwi - with a tournament high 17 offloads - has the chance of becoming the first player to win the World Cup in both rugby league and rugby union after triumphing with the All Blacks in 2011."I'd give every award back in a heartbeat just to get across the line this weekend," said Williams.The (lack of) air up there Watch mCayman Islands-based Webb, the head of Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's Premier League match at Chelsea on Sunday."I am going to be at the match tomorrow and I have asked to meet Yaya Toure," he told BBC Sport."For me it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's complaint. In a statement the Russian side added: "We found no racist insults from fans of CSKA." Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen as early as Sunda.The weather plays an important role in this mission. Starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.The balloon will slowly drift to the edge of space at 120,000 feet ( Then, I would assume, he will slowly step out onto something resembling an Olympic diving platform.They blew it in 2008 when they got caught cold in the final and they will not make the same mistake against the Kiwis in Manchester.Five years ago they cruised through to the final and so far history has repeated itself here - the last try they conceded was scored by England's Josh Charnley in the opening game of the tournament.That could be classed as a weakness, a team under-cooked - but I have been impressed by the Kangaroos' focus in their games since then.They have been concentrating on the sort of stuff that wins you tough, even contests - strong defence, especially on their own goal-line, completing sets and a good kick-chase. They've been great at all the unglamorous stuff that often goes unnoticed in the stands but not by your team-mates.It is as though their entire tournament has been preparation for the final.In Johnathan Thurston, Cooper Cronk, Cameron Smith and either Billy Slater or Greg Inglis at full-back they have a spine that is unmatched in rugby league. They have played in so many high-pressure games - a priceless asset going into Saturday.The Kiwis are a lot less experienced but winning a dramatic match like their semi-final against England will do wonders for their confidence.They defeated Australia in the Four Nations final in 2010 and the last World Cup, and know they can rise to the big occasion.