XAT 2012 Quant : Mensuration - Volume, Surface Area

Mensuration Question 17 - 1 Mark

Ram a farmer, managed to grow shaped-watermelons inside glass cases of different shapes. The shapes he used were: a perfect cube, hemi-spherical, cuboid, cylindrical along with the normal spherical shaped watermelons. Thickness of the skin was same for all the shapes. Each of the glass cases was so designed that the total volume and the weight of the all the water-melons would be equal irrespective of the shape.

A customer wants to buy water-melon for making juice, for which the skin of the water-melon has to be peeled off, and therefore is a waste. Which shape should the customer buy?

1. Cube
2. Hemi-sphere
3. Cuboid
4. Cylinder
5. Normal spherical
   
   

Correct Answer : Normal spherical. Choice (E)

Explanatory Answer

This is the kind of question that is either very easy or very difficult depending on whether you know the concept behind the question.

For a given surface area, the volume contained increases with increasing symmetry of the object. For instance, if we are to make water melons of different shapes of the same surface area, the volume will be maximum when it is made into a sphere.

The corollary is that for a given volume, the surface area will be minimum when the object is a sphere. So, the customer should opt for spherical shaped water melons if she has to minimize wastage.

For 2-dimensional object, for a given perimeter, the area increases with increasing symmetry.

Among different triangles of a given perimeter, an equilateral triangle has the largest area.

The area increases with increasing number of sides - i.e., for a given perimeter the area of a square will larger than that of an equilateral triangle; the area of a regular pentagon of a given perimeter will be larger than that of a square and so on.

Among different regular polygons of a given perimeter / circumference a circle has the largest area.

Level of difficulty : Moderate

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