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