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If the sphere has the same curved surface area as total surface area of cone of vertical height $40 cm$ and radius $30 cm$ , then the radius of the sphere is:

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10 \sqrt{6} cm

10 \sqrt{3} cm

10 \sqrt{2} cm

12 cm

answer is A.

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Detailed Solution

Given,
Height of cone

h = 40 cm

Radius of cone

r = 30 cm

Total surface area of cone can be given as,
Description: C:UsersadminDesktopeiuru34yty54i56.PNG

= πr (\sqrt{r^{2} + h^{2}}) + π r^{2}

= π \times 30 \sqrt{3 0^{2} + 4 0^{2}} + π \times 3 0^{2}

= π \times 30 (\sqrt{900 + 1600} + 30)

= π \times 30 (\sqrt{2500} + 30)

= π \times 30 (50 + 30)

= π \times 30 \times 80

= 2400 π c m^{2} . . . . . . . . . . . . . . (i)

Sphere and cone have same curved surface area. The sphere's curved surface area is

4 π

times than its radius squared. Thus,

4 π (radius)^{2} = 2400 π

= (radius)^{2} = \frac{2400 π}{4 π}

= (radius)^{2} = 600

\Rightarrow radius = \sqrt{600}

= 10 \sqrt{6} cm

.
Hence, the correct option is 1.

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