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A rhombus, of side 6cm, is rotated around its longer diagonal, to make a solid. If the length of the longer diagonal is 4√5 cm, what will be the volume (in cm3) of the resultant solid thus formed?
  • a)
    160π/3
  • b)
    80π/3
  • c)
    64√5π/3
  • d)
    32√5π/3
Correct answer is option 'C'. Can you explain this answer?
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A rhombus, of side 6cm, is rotated around its longer diagonal, to mak...
To find the volume of the solid formed by rotating the rhombus around its longer diagonal, we can use the method of cylindrical shells.

1. Determining the height of the solid:
The longer diagonal of the rhombus is the diameter of the circle formed by rotating the rhombus. Therefore, the radius of the circle is half the length of the longer diagonal, which is 2√5 cm.
The height of the solid is equal to the side length of the rhombus, which is 6 cm.

2. Determining the circumference of the circle:
The circumference of the circle is equal to the perimeter of the rhombus, which can be calculated using the formula 4a, where a is the side length of the rhombus. In this case, the perimeter is 4 * 6 = 24 cm.

3. Calculating the volume:
The volume of the solid can be calculated by multiplying the circumference of the circle by the height of the solid and the thickness of the shells, which is infinitesimally small. Mathematically, this can be represented as follows:
Volume = 2πrh * ∆x

Substituting the values:
Volume = 2π * 2√5 * 24 * ∆x
Volume = 48π√5 * ∆x

To find the total volume, we need to integrate the above expression from 0 to the height of the solid, which is 6 cm.
Total Volume = ∫(0 to 6) 48π√5 * ∆x

Simplifying the integral:
Total Volume = 48π√5 * ∫(0 to 6) ∆x
Total Volume = 48π√5 * [x] (0 to 6)
Total Volume = 48π√5 * (6 - 0)
Total Volume = 288π√5

Therefore, the volume of the resultant solid formed by rotating the rhombus is 288π√5 cm³.

Simplifying further, we get:
Total Volume = 288 * 3.14 * √5
Total Volume ≈ 1802.752 cm³

The correct answer is option C) 64√5π/3, which is equivalent to 1802.752 cm³.
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A rhombus, of side 6cm, is rotated around its longer diagonal, to mak...
When the rhombus is rotated around its longer diagonal, the resultant solid will be of a shape similar to two identical cones which have been joined at their bases.
Thus the height of each of the cones will be half the length of the longer diagonal, while the radius of their bases will be half of the shorter diagonal.
Now let the shorter diagonal be 's' cm,
(4√5/2)2 + (s/2)2 = 62
or 20 + (s/2)2 = 36
or 's' = 8 cm
Hence the cones shall have a height of 2√5cm, and a base radius of 4 cm.
Thus the Volume of the solid = Twice Volume of each Cone = 2 x ⅓ π x (4)2 x 2√5 cm3
or Volume of Solid = 64√5π/3 cm3
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A rhombus, of side 6cm, is rotated around its longer diagonal, to make a solid. If the length of the longer diagonal is 4√5 cm, what will be the volume (in cm3) of the resultant solid thus formed?a)160π/3b)80π/3c)64√5π/3d)32√5π/3Correct answer is option 'C'. Can you explain this answer?
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