The frustum of a solid cone has a top radius of 6 cm and bottom radius of 18 cm. Its height is 10 cm. Calculate: (i) the volume: (ii) the total surface area. (10 marks)

To solve this problem, we need to calculate the volume and the total surface area of the frustum of the cone.<br /><br />(i) Volume:<br />The volume of a frustum of a cone can be calculated using the formula:<br /><br />V = (1/3) * π * h * (r1^2 + r1 * r2 + r2^2)<br /><br />where:<br />V is the volume,<br />h is the height,<br />r1 is the radius of the top base,<br />r2 is the radius of the bottom base.<br /><br />Given:<br />h = 10 cm,<br />r1 = 6 cm,<br />r2 = 18 cm.<br /><br />Substituting the values into the formula, we get:<br /><br />V = (1/3) * π * 10 * (6^2 + 6 * 18 + 18^2)<br />V = (1/3) * π * 10 * (36 + 108 + 324)<br />V = (1/3) * π * 10 * 468<br />V = 1560 * π<br />V ≈ 4905.76 cm^3<br /><br />Therefore, the volume of the frustum of the cone is approximately 4905.76 cm^3.<br /><br />(ii) Total Surface Area:<br />The total surface area of a frustum of a cone can be calculated using the formula:<br /><br />A = π * r1 * r2 * sqrt((r2 - r1)^2 + h^2) + π * (r1 + r2) * sqrt((r1 + r2)^2 - (r1^2 + r2^2))<br /><br />Given:<br />h = 10 cm,<br />r1 = 6 cm,<br />r2 = 18 cm.<br /><br />Substituting the values into the formula, we get:<br /><br />A = π * 6 * 18 * sqrt((18 - 6)^2 + 10^2) + π * (6 + 18) * sqrt((6 + 18)^2 - (6^2 + 18^2))<br />A = π * 6 * 18 * sqrt(12^2 + 10^2) + π * 24 * sqrt(24^2 - (6^2 + 18^2))<br />A = π * 6 * 18 * sqrt(144 + 100) + π * 24 * sqrt(576 - (36 + 324))<br />A = π * 6 * 18 * sqrt(244) + π * 24 * sqrt(576 - 360)<br />A = π * 6 * 18 * 15.620499 + π * 24 * sqrt(216)<br />A = 169.224992 + 165.708504<br />A ≈ 335.933496<br /><br />Therefore, the total surface area of the frustum of the cone is approximately 335.93 cm^2.