Simplify square root of 3 multiplied by the fifth root of 3.
2 answers:
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Answer:
The radius of the sphere is approximately 62.71 cm
Step-by-step explanation:
The question relates to the definition of a sphere
The given parameters are;
The distance between the sharp parallel blades through which the sphere is rolled = 77 centimeters
The radii of the circular faces left by cutting the sphere with the blade = 39 cm and 60 cm
From the triangles formed by the cross-section of the sphere, we have;
AB ║ DE Given
∠A ≅ ∠E, ∠D ≅ ∠B Alternate angles
∠C ≅ ∠C by reflective property
∴ ΔABC ~ ΔDCE by Angle-Angle similarity theorem
CF/CG = AB/DE = 78/120 = 13/20
CF = CG × 13/20
CF + CG = 77
∴ CG × 13/20 + CG = 77
33·CG/20 = 77
∴ CG = 140/3
CF = 13/20 × CG = 13/20 × 140/3 = 91/3
CF = 91/3
The diameter of the sphere AE = AC + EC
By Pythagoras's theorem
AC = √(FA² + CF²) = √(39² + (91/3)²) = 13·√(130)/3
EC = √(CG² + GD²) = √((140/3)² + 60²) = 20·√(130)/3
∴ AE = AC + EC = 13·√(130)/3 + 20·√(130)/3 = 11·√(130)
The diameter, 'D', of the sphere, AE = 11·√(130)
The radius of the sphere = D/2 = 11·√(130)/2 ≈ 62.71 cm
