Bissoy.com
Doctors Medicines MCQs Q&A

Let X be a right circular cylinder with height ‘h’ and diameter ‘d’, Y be a right circular cone with height and diameter equal to that of cylinder X and Z be a sphere of maximum volume that can be inscribed in the cylinder X. Consider the following statements: 1. If d < h, then the volume of Y is (h/2d) times the volume of Z. 2. If d > h, then the volume of Y is (d2/2h2) times the volume of Z. Which of the above statements is/are correct?

Let X be a right circular cylinder with height ‘h’ and diameter ‘d’, Y be a right circular cone with height and diameter equal to that of cylinder X and Z be a sphere of maximum volume that can be inscribed in the cylinder X. Consider the following statements: 1. If d < h, then the volume of Y is (h/2d) times the volume of Z. 2. If d > h, then the volume of Y is (d2/2h2) times the volume of Z. Which of the above statements is/are correct? Correct Answer Both 1 and 2

Given,

Height of cylinder X = Height of cone Y = h

Diameter of cylinder X = Diameter of cone Y = d

As we know,

Volume of cone = (π/3) × (radius)2 × height = (π/12) × (diameter)2 × height

Volume of sphere = (4π/3) × (radius)3 = (π/6) × (diameter)3

Now,

The diameter of the sphere of maximum volume that can be inscribed in a cylinder will be equal to the smaller quantity between the height and the diameter of the cylinder,

Considering statement 1,

If d < h, diameter of sphere Z = d

⇒ Volume of cone Y/Volume of sphere Z = (πd2h/12)/(πd3/6)

⇒ Volume of cone Y/Volume of sphere Z = h/2d

Volume of cone Y is (h/2d) times the volume of sphere Z

Hence, statement 1 is correct

Considering statement 2,

If d > h, diameter of sphere Z = h

⇒ Volume of cone Y/Volume of sphere Z = (πd2h/12)/(πh3/6)

⇒ Volume of cone Y/Volume of sphere Z = d2/2h2

⇒ Volume of cone Y is (d2/2h2) times the volume of sphere Z

Hence, statement 2 is correct

∴ Both statements 1 and 2 are correct

Related Questions

“What is the different between height and radius of solid cylinder?” Which of the following option is sufficient alone to find the answer? A) A solid cylinder is made by melting a solid sphere. The radius of cylinder is 3 times the radius of a circle whose area is 154 sq.m. Height of cylinder is equal to the height of cone whose volume is 2772 cubic meters. B) Area of a rectangle whose longer side is ¾ times the radius of a solid cylinder, is 168 sq.m and, Volume of solid cylinder is 2772 cubic meters and its height is 18 m. C) A metallic cone of radius 15m and height 35m is melted into the solid cylinder of radius equal to the radius of a circle whose area is equal to the area of rectangle whose sides are in the ratio 4 : 3 respectively. D) The curved surface area of the solid cylinder is 3 time the surface area of a sphere of radius 7m and height of cylinder is equal to the height of a metallic cone whose curved surface area is 308 sq.m. 
A cube is inscribed in a sphere. A right circular cylinder is within the cube touching all the vertical faces. A right circular once is inside the cylinder. Their heights are same and the diameter of the cone is equal to that of the cylinder. Consider the following statements: 1) The surface area of the sphere is √5 Times the curved surface area of the cone. 2) The surface area of the cube is equal to the curved surface area of the cylinder. Which of the above statements is/are correct?
Given below are two quantities named A and B. Based on the given information, you have to determine relation between the two quantities. You should use the given data and knowledge of Mathematics to choose between the possible answers. Quantity A: Number of hemispheres that can be formed by melting a sphere having volume equal to the volume of right circular cylinder whose diameteris 6 cm and height is 4 cm. Also, radius of hemisphere formed is half the radius of sphere from which the hemisphere is formed. Quantity B: Number of ice cream cones that can be made from a right circular cylinder shape ice cream container having radius 6 cm and height 5 cm. Ice cream cone to be made should have height 4 cm and slant height 5 cm and there should be a hemisphere made on top of the cone having radius equal to the radius of cone.
A cube is inscribed in a sphere. A right circular cylinder is within the cube touching all the vertical faces. A right circular once is inside the cylinder. Their heights are same and the diameter of the cone is equal to that of the cylinder. What is the ratio of the volume of the sphere to that of cone?
The sum and difference of the slant height and height of cone A are in the ratio 3 ∶ 1, while the sum and difference of the slant height and height of cone Y are in the ratio 5 ∶ 1. If the volume of cone A is 30% of the volume of cone B, then the height of cone A is what percentage of the height of cone B.
 The following question has two statements. Study the question and the statements and decide which of the statement(s) is/are necessary to answer the question.  The height of a right circular cone is ‘h’ and radius is ‘r’. A small cone is cut-off at the top by a plane parallel to the base. At what height above the base, the section has been made? I) Height of cone (h) = 40 cm II) Volume of smaller cone ∶ volume of larger cone = 1 ∶ 30
A cube is inscribed in a sphere. A right circular cylinder is within the cube touching all the vertical faces. A right circular once is inside the cylinder. Their heights are same and the diameter of the cone is equal to that of the cylinder. What is the ratio of the volume of the cube to that of the cylinder ?
The radius of the base of a right circular cone and a sphere is each equal. If the volume of sphere and cone is same, then what is the ratio of height of cone to the radius of sphere.
A solid sphere of diameter 18 cm is melted and cast into a right circular cone whose base radius is 1/√2 times its slant height. If the radius of the sphere and the cone can are equal, then how many such cones can be made out of the sphere? 
The radius of a sphere is equal to the radius of a cone and the volume of the sphere is equal to the volume of the cone. Find the curved surface area of the sphere when the height of the cone is 28 m. (Take π = 22/7)