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【题目】某学校要建造一个面积为10000平方米的运动场如图,运动场是由一个矩形ABCD和分别以AD、BC为直径的两个半圆组成跑道是一条宽8米的塑胶跑道,运动场除跑道外,其他地方均铺设草皮已知塑胶跑道每平方米造价为150元,草皮每平方米造价为30元

1设半圆的半径OA=r,设建立塑胶跑道面积S与r的函数关系Sr

2由于条件限制r∈[30,40],问当r取何值时,运动场造价最低?最低造价为多少?精确到元

【答案】见解析

【解析】1塑胶跑道面积S=π[r2r-82]+8××2=+8πr-64π

∵πr2<10000,∴0<r<

1设运动场的造价为y元,y=150×+30×

=300000+120×7680π

fr+8πr,∵f′r=8π当r∈[30,40]时,f′r<0,

∴函数y=300000+120×-7680π在[30,40]上为减函数

∴当r=40时,ymin≈636510,即运动场的造价最低为636510元

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