We shall show that the closed sphere with centre (2,3,7) and radius 10 in R3 is contained in the
open cube P={(x,y,z):x−2<11,y−3<11,z−7<11}.
The equation of the sphere with centre (2, 3, 7) and radius 10 in R3 is
(x−2)2+(y−3)2+(z−7)2=102(x−2)2≥0,x∈R(y−3)2≥0,y∈R(z−7)2≥0,z∈R
Then
0≤(x−2)2≤102∣x−2∣≤100≤(y−3)2≤102∣y−3∣≤100≤(z−7)2≤102∣z−7∣≤10
Hence
x−2<11,y−3<11,z−7<11,x,y,z∈R
This means that the closed sphere with centre (2,3,7) and radius 10 in R3 is contained in the
open cube P={(x,y,z):x−2<11,y−3<11,z−7<11}.
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