Question

A circus tent is in the form of a cone over a cylinder. The
diameter of the base is 6 m, the height of the cylindrical part is
15 m and the total height of the tent is 19 m. The canvas
required for the tent is
A: 110pi m
B: 128pi m
C: 105pi m
D: 125pi m​

Answer 1

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

Dimensions of cylinder

Diameter of base = 6 m

So,

• Radius of cylinder, r = 3 m

• Height of cylindrical part, h = 15 m

So, Amount of canvas required to make cylindrical part = Curved Surface Area of cylinder.

We know,

Curved Surface Area of cylinder is

$\bf :\longmapsto\:CSA_{(cylinder)} = 2\pi \: r \: h$

$\rm \: = \: \:2\pi \: \times 3 \times 15$

$\rm \: = \: \:90\pi \:$

$\bf :\longmapsto\:CSA_{(cylinder)} = 90 \: \pi \: {m}^{2}$

Dimensions of cone

Diameter of base = 6 m

So,

Radius of cone, r = 3 m

Height of conical part, H = 19 - 15 = 4 m

So, Amount of canvas required to make conical part = Curved Surface Area of cone.

We know,

Curved Surface Area of cone is

$\rm :\longmapsto\:CSA_{(cone)} = \pi \: r \: l$

where,

• l is slant height.

We know,

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

$\rm :\longmapsto\:l = \sqrt{ {4}^{2} + {3}^{2} }$

$\rm :\longmapsto\:l = \sqrt{ 16+ 9 }$

$\rm :\longmapsto\:l = \sqrt{ 25 }$

$\rm :\longmapsto\:l = 5 \: m$

Hence,

Curved Surface Area of cone is

$\rm :\longmapsto\:CSA_{(cone)} = \pi \: r \: l$

$\rm :\longmapsto\:CSA_{(cone)} = \pi \: \times 3\: \times 5$

$\bf :\longmapsto\:CSA_{(cone)} =15 \pi \: {m}^{2}$

Hence,

Amount of Canvas required to make a tent is

$\bf :\longmapsto\:Amount_{(canvas \: required)}$

$\rm \: = \: \:CSA_{(cylinder)} + CSA_{(cone)}$

$\rm \: = \: \:90\pi \: + \: 15\pi$

$\rm \: = \: \:105\pi \:$

$\bf\implies \:Amount_{(canvas \: required)} = 105 \: \pi \: {m}^{2}$

So,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf{ \: Option \: C \: is \: correct}}}$

Additional Information :-

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