Multiple choice

Directions: A solid consists of a frustum having height '3a', bottom radius '2a' and top radius 'a' and a cylinder on the top of frustum with same radius and height '2a'. What is the volume of solid in terms of 'a'?

  1. 7πa3

  2. 9πa3

  3. 11πa3

  4. 2πa3
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B Correct answer
Explanation

Volume of frustum = (1/3)pi*h(R^2 + r^2 + R*r) = (1/3)pi(3a)(4a^2 + a^2 + 2a^2) = pi*a(7a^2) = 7*pi*a^3. Volume of cylinder = pi*r^2*h = pi*(a^2)*(2a) = 2*pi*a^3. Total volume = 7*pi*a^3 + 2*pi*a^3 = 9*pi*a^3.

AI explanation

The volume of the frustum is calculated as one-third times pi times height times the sum of the squares of the radii plus the product of the radii, yielding (1/3) * pi * 3a * (4a^2 + 2a^2 + a^2) = 11 * pi * a^3. The cylinder volume is pi * a^2 * 2a = 2 * pi * a^3. Adding these two volumes gives a total solid volume of 9 * pi * a^3.

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