Multiple choice

Fill in the blank: The volume of a frustum cone top and bottom area of a circle and height is given. Choose the formula which satisfy the given statement ________________.($A_1, A_2 , h$ are area of top circle, area of bottom circle and height of frustum respectively).

  1. $\frac{1}{3} (A_1 + A_2 + \sqrt{A_1 A_2 }) h$
  2. $\frac{1}{3} (A_1 - A_2 + \sqrt{A_1 A_2 }) h$
  3. $\frac{1}{3} (A_1 + A_2 - \sqrt{A_1 A_2 }) h$
  4. $\frac{2}{3} (A_1 + A_2 + \sqrt{A_1 A_2 }) h$
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A Correct answer
Explanation

The volume of a frustum of a cone is V = (1/3) * pi * h * (r1^2 + r2^2 + r1*r2). Since A = pi * r^2, then r^2 = A/pi and r = sqrt(A/pi). Substituting these, V = (1/3) * h * (A1 + A2 + sqrt(A1*A2)).

AI explanation

The volume of a frustum of a cone can be expressed in terms of its top area, bottom area, and height. The formula is derived from the standard volume formula involving radii, V = (1/3) * pi * (R^2 + R*r + r^2) * h. By substituting the areas A1 = pi*R^2, A2 = pi*r^2, and square root of (A1*A2) = pi*R*r, the volume becomes (1/3) * (A1 + A2 + square root of (A1*A2)) * h. This matches the required algebraic expression.

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