Question

Schel.
43.
The base radius and height of a cylindrical block of wood are 8 centimetres and 15
centimetres. A cone of maximum size is carved out of this.
(a) What are the radius and height of the cone?
(b)
Find its slant height.
(c)
Find the curved surface area of this cone.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

• Height of cylindrical block, h = 15 cm

and

• Radius of cylindrical block, r = 8 cm

According to statement,

• A cone of maximum size is carved out from this cylindrical block.

Therefore,

$\rm :\longmapsto\:Radius_{(cone)} = Radius_{(cylinder)} = 8 \: cm$

$\rm :\longmapsto\:Height_{(cone)} = Height_{(cylinder)} = 15 \: cm$

So,

Dimensions of Cone are

• Radius of cone, r = 8 cm

• Height of cone, h = 15 cm

Now,

We know that

• Slant height (l) of cone is

$\rm :\longmapsto\:l \: = \sqrt{ {r}^{2} + {h}^{2} }$

$\rm :\longmapsto\:l \: = \sqrt{ {8}^{2} + {15}^{2} }$

$\rm :\longmapsto\:l = \sqrt{64 + 225}$

$\rm :\longmapsto\:l = \sqrt{289}$

$\bf\implies \: \boxed{ \bf{l \: = \: 17 \: cm}}$

$\bf\implies \: \boxed{ \bf{Slant \: height \: of \: cone, \: l \: = \: 17 \: cm}}$

Now,

We know that,

$\rm :\longmapsto\:\bf \: Curved \: Surface \: Area_{(cone)} = \pi \: rl$

$\rm :\longmapsto\:Curved \: Surface \: Area_{(cone)} = \dfrac{22}{7} \times 8 \times 17$

$\rm :\longmapsto\:Curved \: Surface \: Area_{(cone)} = \dfrac{2992}{7} \: {cm}^{2}$

Additional Information :-

Perimeter of rectangle = 2(length× breadth)

Diagonal of rectangle = √(length²+breadth²)

Area of square = side²

Perimeter of square = 4× side

Volume of cylinder = πr²h

T.S.A of cylinder = 2πrh + 2πr²

Volume of cone = ⅓ πr²h

C.S.A of cone = πrl

T.S.A of cone = πrl + πr²

Volume of cuboid = l × b × h

C.S.A of cuboid = 2(l + b)h

T.S.A of cuboid = 2(lb + bh + lh)

C.S.A of cube = 4a²

T.S.A of cube = 6a²

Volume of cube = a³

Volume of sphere = 4/3πr³

Surface area of sphere = 4πr²

Volume of hemisphere = ⅔ πr³

C.S.A of hemisphere = 2πr²

T.S.A of hemisphere = 3πr²