Question

38. A solid wooden toy is in the form of a hemisphere surrounded by a cone of same radius. The
radius of hemisphere is 3.5 cm and the total wood used in making of toy is 166 cm3. Find the
height of toy. Also find the cost of painting the hemisphere part of the toy at the rate of Rs. 10 per
cm.​

Answer 1

Answer:

1. Height of the toy = 9.5 cm
2. Cost of painting the hemispherical part of the toy = ₹ 770

Step-by-step explanation:

Given,

• Radius of Hemisphere = Radius of Cone = 3.5 cm
• Volume of wood used in making toy = 166 ⁵/₆ cm³ = 1001/6 cm³

Let the height of the Cone be 'h'.

⇒ Volume of wood = Volume of hemisphere + Volume of cone

                                = [²/₃ × $\pi$ × r³] + [¹/₃ ×

                                = [¹/₃ × $\pi$ × r²] × [2×r + h]

⇒ ¹⁰⁰¹/₆                     = [¹/₃ × ²²/₇ × 3.5 × 3.5] × [2×3.5 + h]

⇒ [2×3.5 + h]           = [¹⁰⁰¹/₆] ÷ [¹/₃ × ²²/₇ × 3.5 × 3.5]

⇒ 7 + h                    =  ¹⁰⁰¹/₆ × 3 × ⁷/₂₂ × ¹/₃.₅ × ¹/₃.₅

                                = ²¹⁰²¹/₁₆₁₇

⇒ 7 + h                    = 13 cm

⇒ ∴ h                      = 13 - 7

                               = 6 cm

Height of the cone = 6 cm

Height of the toy = Height of cone + Height of hemisphere

                            = 6 cm + 3.5 cm     [∵Ht. of hemisphere = Its Radius]

                            = 9.5 cm

The part of the hemisphere which would be painted is the CURVED SURFACE AREA of the hemisphere.

Curved surface area of hemisphere = 2 × $\pi$ × r²

                                                            = 2 × ²²/₇ × 3.5 × 3.5

                                                            = 77 cm²

Rate of painting the hemispherical part of the toy = ₹ 10 per m²

Cost of painting the hemispherical part of the toy = 77 × 10

                                                                                   = ₹ 770

Hence, the Height of the toy is 9.5 cm & Cost of painting the hemispherical part of the toy is  ₹ 770.

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