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  1. Chapter 13 Class 10 Surface Areas and Volumes
  2. Serial order wise
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Ex 13.5, 6 Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained. There are two cones OCD & OAB We are given Height of frustum = h Slant height of frustum = l Radius PB = r1 Radius QD = r2 We need to find Curved Surface Area & Total Surface Area Here, We need to write h1, l1, h2, l2 in terms of h and l Curved Surface area of frustum ABDC = Curved Surface area of cone OAB – Curved Surface area of cone OCD = πr1l1 – πr2l2 In Δ OPB & Δ OQD ∠ BOP = ∠ DOQ ∠ OPB = ∠ OQD So, Δ OPB ∼ Δ OQD ∴ 𝑃𝐵/𝑄𝐷 = 𝑂𝐵/𝑂𝐷 𝑟_1/𝑟_2 = 𝑙_1/𝑙_2 Putting l1 = l + l2 𝑟_1/𝑟_2 = (𝑙 + 𝑙_2)/𝑙_2 𝑟_1/𝑟_2 = 𝑙/𝑙_2 + 𝑙_2/𝑙_2 𝑟_1/𝑟_2 = 𝑙/𝑙_2 + 1 𝑟_1/𝑟_2 −1 = 𝑙/𝑙_2 (𝑟_1 − 𝑟_2)/𝑟_2 = 𝑙/𝑙_2 𝑙_2 ((𝑟_1 − 𝑟_2)/𝑟_2 ) = 𝑙 𝑙_2 = 𝑙(𝑟_2/(𝑟_1 − 𝑟_2 )) From (1) Curved Surface area of frustum = πr1l1 – πr2l2 Putting l1 = l + l2 = πr1(l + l2) – πr2l2 From (2): Putting 𝑙_2 = 𝑙(𝑟_2/(𝑟_1 − 𝑟_2 )) = πr1(l + 𝑙(𝑟_2/(𝑟_1 − 𝑟_2 ))) – πr2𝑙(𝑟_2/(𝑟_1 − 𝑟_2 )) = πr1l (1 + (𝑟_2/(𝑟_1 − 𝑟_2 ))) – πr2𝑙(𝑟_2/(𝑟_1 − 𝑟_2 )) = πr1l ((𝑟_1 − 𝑟_2 + 𝑟_2)/(𝑟_1 − 𝑟_2 )) – πr2𝑙(𝑟_2/(𝑟_1 − 𝑟_2 )) = πr1l (𝑟_1/(𝑟_1 − 𝑟_2 )) – πr2𝑙(𝑟_2/(𝑟_1 − 𝑟_2 )) = πl (〖𝑟_1〗^2/(𝑟_1 − 𝑟_2 )) – π𝑙(〖𝑟_2〗^2/(𝑟_1 − 𝑟_2 )) = πl ((〖𝑟_1〗^2− 〖𝑟_2〗^2)/(𝑟_1 − 𝑟_2 )) = πl (((𝑟_1 − 𝑟_2)(𝑟_1+ 𝑟_2))/(𝑟_1 − 𝑟_2 )) = πl(𝑟_1+ 𝑟_2) = π(𝑟_1+ 𝑟_2 )𝑙 ∴ Curved Surface area of frustum = π(𝒓_𝟏+ 𝒓_𝟐 )𝒍 Total Surface area of frustum = Curved Surface area of frustum + Area of top circle + Area of bottom circle = π(𝑟_1+ 𝑟_2 )𝑙 + π〖𝑟_1〗^2 + π〖𝑟_2〗^2 ∴ Total Surface area of frustum = π(𝒓_𝟏+ 𝒓_𝟐 )𝒍 + π〖𝒓_𝟏〗^𝟐 + π〖𝒓_𝟐〗^𝟐

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