Question

（9分）
操作：小明準(zhǔn)備制作棱長(zhǎng)為1cm的正方體紙盒，現(xiàn)選用一些廢棄的圓形紙片進(jìn)行如下設(shè)計(jì)：
 

紙片利用率=×100%
發(fā)現(xiàn)：（1）方案一中的點(diǎn)A、B恰好為該圓一直徑的兩個(gè)端點(diǎn)．你認(rèn)為小明的這個(gè)發(fā)現(xiàn)是否正確，請(qǐng)說(shuō)明理由．
（2）小明通過(guò)計(jì)算，發(fā)現(xiàn)方案一中紙片的利用率僅約為38.2%．請(qǐng)幫忙計(jì)算方案二的利用率，并寫(xiě)出求解過(guò)程．
探究：（3）小明感覺(jué)上面兩個(gè)方案的利用率均偏低，又進(jìn)行了新的設(shè)計(jì)（方案三），請(qǐng)直
接寫(xiě)出方案三的利用率．

Answer 1

發(fā)現(xiàn)：（1）小明的這個(gè)發(fā)現(xiàn)正確．····················································· 1分
理由：解法一：如圖一：

連接AC、BC、AB，∵AC＝BC＝，AB＝
∴AC²+BC²＝AB²   ∴∠BAC＝90°，····························································· 2分
∴AB為該圓的直徑．················································································ 3分
解法二：如圖二：

連接AC、BC、AB．易證△AMC≌△BNC，∴∠ACM＝∠CBN．
又∵∠BCN＋∠CBN＝90°，∴∠BCN＋∠ACM＝90°，即∠BAC＝90°，·················· 2分
∴AB為該圓的直徑．················································································ 3分
（2）如圖三：

易證△ADE≌△EHF，∴AD＝EH＝1．·························································· 4分
∵DE∥BC，∴△ADE∽△ACB，∴＝∴＝，∴BC＝8．·················· 5分
∴S_△ACB＝16．························································································ 6分
∴該方案紙片利用率＝×100%＝×100%＝37.5%···················· 7分
探究：（3）······················································································· 9分

略