21、
(1) A(8,0)、P(x,y) (x>0、y>0)
x+y=10
y=10-x
∵x>0、y>0
∴10-x>0
x<10
0<x<10
S=1/2×8y
=4y
=4(10-x)
=40-4x
即:S=40-4x
定义域:0<x<10
(2) y=4x
10-x=4x
5x=10
x=2
S=40-4×2
=32
(3) 周长=OA+AP+OP
=8+√[(x-8)^2+y^2]+√(x^2+y^2)
=8+√[x^2-16x+64+(10-x)^2]+√[x^2+(10-x)^2]
=8+√(2x^2-36x+164)+√(2x^2-20x+100)
=8+√2{√[(x-9)^2+1]+√[(x-5)^2+25]}
对于√[(x-9)^2+1]+√[(x-5)^2+25],当√[(x-9)^2+1]=√[(x-5)^2+25]时取得最小值
(x-9)^2+1=(x-5)^2+25
(x-9)^2-(x-5)^2-24=0
[(x-9)+(x-5)][(x-9)-(x-5)]-24=0
(2x-14)(-4)-24=0
-8x+56-24=0
8x=32
x=4
即当x=4时,周长最小为:
8+√2{√[(4-9)^2+1]+√[(4-5)^2+25]}
=8+√2[√26+√26]
=8+2√52
=8+4√13
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