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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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Most Upvoted Answer
A rhombus, of side 6cm, is rotated around its longer diagonal, to mak...
To find the volume of the solid obtained by rotating the rhombus around its longer diagonal, we can use the formula for the volume of a solid of revolution.

The formula for the volume of a solid of revolution is given by V = π∫(y²)dx, where y is the distance from the axis of rotation to the function and x is the variable of integration.

Let's consider a coordinate system where the longer diagonal of the rhombus lies along the x-axis and the shorter diagonal lies along the y-axis. The coordinates of the vertices of the rhombus can be taken as (-3, 0), (3, 0), (0, -2), and (0, 2).

- Identify the Function
The function that represents the upper half of the rhombus is y = 2 - (2/3)x. This function represents the distance from the x-axis to the upper half of the rhombus.

- Identify the Axis of Rotation
The longer diagonal of the rhombus, which lies along the x-axis, will be the axis of rotation.

- Set up the Integral
We need to find the limits of integration for the integral. Since the rhombus has a side length of 6 cm and the longer diagonal has a length of 4√5 cm, we can use similar triangles to find the length of the shorter diagonal.

The ratio of the lengths of the shorter diagonal and the longer diagonal is equal to the ratio of the sides opposite these diagonals in the rhombus. Therefore, (2/3)x / 6 = 2 / (4√5).

Simplifying this equation, we get x = 3√5.

So, the limits of integration for the integral will be from -3√5 to 3√5.

- Evaluate the Integral
Substituting the function and the limits of integration into the integral, we get:

V = π∫[2 - (2/3)x]²dx, from -3√5 to 3√5.

Simplifying and evaluating this integral will give us the volume of the solid.

- Calculate the Volume
The integral evaluates to V = π[(8/45)x³ - (4/3)x] from -3√5 to 3√5.

Substituting the limits of integration and simplifying, we get:

V = π[(8/45)(3√5)³ - (4/3)(3√5)] - π[(8/45)(-3√5)³ - (4/3)(-3√5)]

V = π(8√5/45)(135 - 15) + π(8√5/45)(135 + 15)

V = π(8√5/45)(2 * 135)

V = π(8√5/45)(270)

V = (64√5π/3).

Therefore, the volume of the solid obtained by rotating the rhombus around its longer diagonal is 64√5π/3 cm³. Hence, option C is the correct answer.
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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*1/3*π*(4)2 *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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