12.3 Cylinders and Cones: The Rotation Solids

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Key Formulas

Solved Examples

Step 1

  1. Decompose Solid: The toy consists of a Cone (top) + Hemisphere (bottom). The flat bases are glued together, so they are internal. We only need CSAs.

Step 2

  1. Dimensions: Radius r=3.5r = 3.5 cm. Total Height H=15.5H = 15.5 cm. Height of hemisphere part = radius = 3.5 cm. Height of cone part h=H−r=15.5−3.5=12h = H - r = 15.5 - 3.5 = 12 cm.

Step 3

  1. Slant Height: l=h2+r2=122+3.52=144+12.25=156.25=12.5l = \sqrt{h^2 + r^2} = \sqrt{12^2 + 3.5^2} = \sqrt{144 + 12.25} = \sqrt{156.25} = 12.5 cm.

Step 4

  1. Calculate Area: TSA of Toy = CSA(Cone) + CSA(Hemisphere).

Step 5

  1. Formula: πrl+2πr2=πr(l+2r)\pi rl + 2\pi r^2 = \pi r(l + 2r).

Step 6

  1. Substitute: 227×3.5(12.5+2(3.5))=11×(12.5+7)=11×19.5=214.5\frac{22}{7} \times 3.5 (12.5 + 2(3.5)) = 11 \times (12.5 + 7) = 11 \times 19.5 = 214.5 cm².
Final Answer

214.5 cm²

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