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x²+h²=c²
x²+h²=10²
h²=100-x²
h=√(100-x²)
P=1/2*(a+b)*h
P(x)=1/2*(10+2x+10)*√(100-x²) , x∈(0,10)
P(x)=(10+x)*√(100-x²)
P(x)=√[(100-x²)(10+x)²]=√[(100-x²)(100+20x+x²)]=
√(10000+2000x+100x²-100x²-20x³-x⁴)=
√(-x⁴-20x³+2000x+10000)
Wyznaczam x, dla ktorego wyrazenie pod pierwiastkiem ma wartosc najwieksza.
(-x⁴-20x³+2000x+10000)'=-4x³-60x²+2000=0 /:(-4)
x³+15x²-500=0
x³-5x²+20x²-500=0
x²(x-5)+20(x²-25)=0
x²(x-5)+20(x+5)(x-5)=0
(x-5)(x²+20x+100)=0
(x-5)(x+10)²=0
x=5 v x=-10∉D
a=10+2*5=10+10=20
Odp. a=20.