Multiple choice

The surface area of a cube of length x is equal to the surface area of a sphere of radius y. Consider the following statements: 1. 2x > 3y 2. The volume of the cube is greater than the volume of the sphere. Which of the above statements is/are correct?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Surface area of cube = 6x^2. Surface area of sphere = 4 * pi * y^2. 6x^2 = 4 * pi * y^2 implies x^2 = (2/3) * pi * y^2, so x = y * sqrt(2 * pi / 3) approx 1.447y. Thus 2x approx 2.89y, so 2x > 3y is false. Volume of cube = x^3 approx 3.03y^3. Volume of sphere = (4/3) * pi * y^3 approx 4.18y^3. So volume of cube is not greater than volume of sphere. Both statements are false.

AI explanation

Equating the surface areas using the formulas 6x squared and 4 pi y squared gives the relation x squared equals two thirds pi y squared, so x equals the square root of two thirds pi multiplied by y. Two times the square root of two thirds pi is less than 3, making the statement 2x greater than 3y incorrect. The volume of the cube is x cubed, which equals two thirds pi y cubed, making it less than the sphere volume of four thirds pi y cubed, so neither statement is correct.

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