Question

The surface area of a sphere of radius 5 cm is five times the
area of the curved surface of a cone of radius 4 cm. Find the height and the volume of
the cone

Answer 1

Answer:

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Step-by-step explanation:

Surface area of the sphere = 4π × 5 × 5 cm²

Curved surface area of the cone = π × 4 × l cm²,

where l is the slant height of the cone.  

According to the statement  

4π × 5 × 5 = 5 × π × 4 × l  

or l = 5 cm.  

Now, l² = h² + r²  

Therefore, (5)² = h² + (4)²  

where h is the height of the cone

or (5)² – (4)² = h²  

or (5 + 4) (5 – 4) = h²

or 9 = h²

or h = 3 cm  

Volume of Cone = 1/3 πr² h  

= 1/3 x 22/7 x 4 x 4 x 3 cm³

= (22 x 16)/7 cm³

= 352/7 cm³

= 50.29 cm³

Answer 2

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☆Required Answer☆

Surface area of sphere

$\small \rightarrow4\pi r ^{2} = 4\pi \times 5 \times 5 = 100\pi cm^{2}$

Curved Surface area of cone

$\rightarrow\pi r1cm^{2} = 4\pi rcm^{2}$

$\rightarrow∴100\pi = 5(4\pi1) \\ \\ \implies1 = 5cm \\ \\ now \: {1}^{2} = {h}^{2} + {r}^{2} \\ \\ \implies {5}^{2} = {h}^{2} + {4}^{2} \\ \\ \implies {h}^{2} = 9 \\ \\ h = 3$

Volume of a cone

$\rightarrow \frac{1}{3} r ^{2} \pi h \\ \\ = \frac{1}{3} \times \frac{22}{7} \times 4 \times 4 \times 3 \\ \\ = \frac{35.2}{7} = 50.29cm^{3}$

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