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A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. If the total height of the toy is 15.5 cm, find the total surface area of the toy.

Mensuration

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Answer

The figure of the toy in the form of a cone surmounted on a hemisphere of same radius is shown below:

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. If the total height of the toy is 15.5 cm, find the total surface area of the toy. Mensuration, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Total height of the toy = 15.5 cm

Radius of the base of the conical part (r) = 3.5 cm.

Height of the cone = 15.5 - 3.5 = 12 cm.

Slant height of the cone = l.

l = r2+h2\sqrt{r^2 + h^2}

l=3.52+122=12.25+144=156.25=12.5 cm.l = \sqrt{3.5^2 + 12^2} \\[1em] = \sqrt{12.25 + 144} \\[1em] = \sqrt{156.25} \\[1em] = 12.5 \text{ cm}.

Total surface area of the toy (T) = Curved surface area of cone + Curved surface area of hemisphere.

∴T=πrl+2πr2=πr(l+2r)=227×3.5×(12.5+2×3.5)=227×3.5×19.5=22×0.5×19.5=214.5 cm2.\therefore T = πrl + 2πr^2 \\[1em] = πr(l + 2r) \\[1em] = \dfrac{22}{7} \times 3.5 \times (12.5 + 2 \times 3.5) \\[1em] = \dfrac{22}{7} \times 3.5 \times 19.5 \\[1em] = 22 \times 0.5 \times 19.5 \\[1em] = 214.5 \text{ cm}^2.

Hence, the total surface area of the toy is 214.5 cm2.

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