Multiple choice

Directions : Answer the questions independently of each other. A cube is completely filled with water. 50% of the water in the cube is used to fill 90% of the volume of a sphere of radius Rs. 75% of the remaining water in the cube fills 60% of the volume in a cone with its height equal to its base radius Ro and the balance water left in the cube after this is used to fill 30% of a cylinder with its height equal to one-third its base radius Rc. Then, Rs3: Ro3: Rc3 =

  1. 3 : 15 : 10

  2. 10 : 27 : 30

  3. 10 : 45 : 9

  4. 2 : 9 : 6

  5. 5
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let cube volume be V. 0.5V = 0.9 * (4/3)pi*Rs^3. 0.75 * 0.5V = 0.6 * (1/3)pi*Ro^3. 0.125V = 0.3 * pi*Rc^2*(Rc/3) = 0.1 * pi*Rc^3. Solving for ratios of cubes leads to 2:9:6.

AI explanation

Let the total volume of the cube be V. From the first condition, 50% of V equals 90% of the sphere's volume, so (1/2)V = 0.9(4/3)pi*Rs^3, which simplifies to Rs^3 = 5*V/(12*pi). The remaining water is V/2, and 75% of it fills 60% of the cone, meaning (3/8)*V = 0.6(1/3)pi*Ro^3, yielding Ro^3 = 15*V/(4.8*pi). The final remaining water is V/8, which fills 30% of the cylinder, so (1/8)*V = 0.3*pi*Rc^2(Rc/3), yielding Rc^3 = 5*V/(4*pi). Finding the ratio of Rs^3 : Ro^3 : Rc^3 gives (5/12) : (15/4.8) : (5/4), which simplifies to 2 : 9 : 6.

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