Question 34 (b) - CBSE Class 10 Sample Paper for 2027 Boards - Maths Standard - Solutions of Sample Papers for Class 10 Boards

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Question 34 (b) A solid wooden toy is in the form of a hemisphere surmounted by a cone of the same radius. The radius of hemisphere is 3.5 cm and the total wood used in making the toy is 166(5)/(6) cm^(3). Find the height of the conical part. Also, find the cost of painting the hemispherical part of the toy at the rate of ₹ 15 per cm^(2). Given Radius of hemisphere = r = 3.5 cm Let height of cone = h And, Total wood used to make toy = 166(5)/(6) cm^(3) = 166+ (5)/(6) cm^(3) = (166 × 6 + 5)/(6) cm^(3) = (996 + 5)/(6) cm^(3) = (𝟏𝟎𝟎𝟏)/(𝟔) 𝒄𝒎^(𝟑) Now, Total wood used to make toy = Total volume of solid (𝟏𝟎𝟎𝟏)/(𝟔) = Volume of cone + Volume of hemisphere Putting values (𝟏𝟎𝟎𝟏)/(𝟔)=(𝟏)/(𝟑)𝝅𝒓𝟐𝒉+(𝟐)/(𝟑)𝝅𝒓^(𝟑) (1001)/(6)=(1)/(3)𝜋𝑟^(2)(ℎ+2𝑟) (1001)/(6)=(1)/(3) ×(22)/(7) × (3.5)^(2) (ℎ+2 × 3.5) (1001)/(6)=(1)/(3) ×(22)/(7) × ((7)/(2))^(2) (ℎ+7) (1001)/(6)=(1)/(3) ×(22)/(7) ×(49)/(4) (ℎ+7) (1001)/(6)=(1)/(3) ×(22)/(7) ×(49)/(4) (ℎ+7) r = 3.5 cm h = ? = (166 × 6 + 5)/(6) cm^(3) = (996 + 5)/(6) cm^(3) = (𝟏𝟎𝟎𝟏)/(𝟔) 𝒄𝒎^(𝟑) Now, Total wood used to make toy = Total volume of solid (𝟏𝟎𝟎𝟏)/(𝟔) = Volume of cone + Volume of hemisphere Putting values (𝟏𝟎𝟎𝟏)/(𝟔)=(𝟏)/(𝟑)𝝅𝒓𝟐𝒉+(𝟐)/(𝟑)𝝅𝒓^(𝟑) (1001)/(6)=(1)/(3)𝜋𝑟^(2)(ℎ+2𝑟) (1001)/(6)=(1)/(3) ×(22)/(7) × (3.5)^(2) (ℎ+2 × 3.5) (𝟏𝟎𝟎𝟏)/(𝟔)=(𝟏)/(𝟑) ×(𝟐𝟐)/(𝟕) × ((𝟕)/(𝟐))^(𝟐) (𝒉+𝟕) (1001)/(6)=(1)/(3) ×(22)/(7) ×(49)/(4) (ℎ+7) (1001)/(6)=(1)/(3) ×(22)/(7) ×(49)/(4) (ℎ+7) (𝟏𝟎𝟎𝟏)/(𝟔)=(𝟏)/(𝟑) ×(𝟐𝟐)/(𝟕) × ((𝟕)/(𝟐))^(𝟐) (𝒉+𝟕) (1001)/(6)=(1)/(3) × 11 ×(7)/(2) (ℎ+7) (1001 × 3 × 2)/(6 × 11 × 7)= (ℎ+7) 13=(ℎ+7) 13 – 7 = h h = 6 cm Thus, height of conical part is 6 cm Now, we are asked Also, find the cost of painting the hemispherical part of the toy at the rate of ₹ 15 per cm^(2). Thus, we need to find Curved Surface area of Hemisphere r = 3.5 cm h = 6 cm Curved surface area of hemisphere = 2𝜋𝑟2 = 2 ×(22)/(7)× (3.5)2 = 2 ×(22)/(7)× (3.5)2 = 2 ×(22)/(7)× 3.5 × 3.5 = 2 × 22 × 0.5 × 3.5 = 77 cm^(2) We need to find the cost of painting the hemispherical part of toy at rate ₹ 15 per cm^(2) Thus, Cost of painting the hemispherical part for 1m^(2) = ₹ 15 Cost of painting the hemispherical part for 77m^(2) = 15 × 77 = Rs 1,155