Question

A solid consists of a cone and a hemisphere which share a common base. The cone has
height of 6cm and base radius 4cm. Find volume and total surface area​

Answer 1

Given :-

• Height of cone = 6cm

• Radius of cone and hemisphere = 4cm

To Find :-

• The volume and total surface area of Solid.

Formulae used:-

• Volume of cone = ⅓πr²h

• Volume of hemisphere = 2/3πr³

• Curved surface area of Cone = πrl

• Curved surface area of hemisphere = 2πr²

How to solve ?

• First we have to find the seperate volume of cone and hemisphere. then we have to add both of them to find the Volume solid

• Proceed same to find the Total surface area of solid but we have to find CSA of both the shape

Now,

→ Volume of cone = ⅓πr²h

→ ⅓ × 22/7 × 4 × 4 × 6

→ 22/7 × 4 × 4 × 2

→ 704/7

→ 100.57cm³

Now,

→ Volume of hemisphere = 2/3πr³

→ 2/3 × 22/7 × 4 × 4 × 4

→ 2816/21

→ 134.09cm³

Therefore,

→ Volume of solids = 100.57 + 134.09

→ Volume of solid = 234.66cm³

Now,

→ We have to find the slant height

→ using Pythogoras thereom

→ r² + h² = l²

→ (4)² + (6)² = l²

→ 16 + 36 = l²

→ 52 = l²

→ l = 7.21cm

Therefore,

→ C. S. A of cone = πrl

→ 22/7 × 4 × 7.21

→ 90.65cm²

Now,

→ C. S. A of hemisphere = 2πr²

→ 2 × 3.14 × 4 × 4

→ 100.48cm²

Therefore,

→ T. S. A of shape = 100.48 + 90.65

→ T. S . A of shape = 191.13cm²

Hence, The Total surface area of solid is 191.13cm²

Answer 2

$\large\underline\blue{\bold{Given \: Question :- }}$

• A solid consists of a cone and a hemisphere which share a common base. The cone has height of 6cm and base radius 4cm. Find volume and total surface area.

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$\huge \orange{AηsωeR} ✍$

$\large\underline\blue{\bold{Given :- }}$

• A solid consists of a cone and a hemisphere which share a common base.
• The cone has height of 6cm and base radius 4cm.

$\large\underline\blue{\bold{To \: Find :- }}$

• Volume of solid
• Total Surface area of solid

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$\large\underline\blue{\bold{Formula \: used :- }}$

${{ \boxed{\large{\bold\green{Surface Area_{(Cone)}\: = \:\pi rl}}}}}$

${{ \boxed{\large{\bold\green{Surface Area_{(hemisphere)}\: = \:2\pi r^2}}}}}$

${{ \boxed{\large{\bold\green{Volume_{(cone)}\: = \dfrac{1}{3} \:\pi r^2h}}}}}$

${{ \boxed{\large{\bold\green{Volume_{(hemisphere)}\: = \dfrac{2}{3} \:\pi r^3}}}}}$

where,

• r = radius of cone
• h = height of cone
• l = slant height of cone

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$\large\underline\blue{\bold{Solution:- }}$

$\bf \:\large \red{AηsωeR : 1.} ✍$

Radius of cone, r = 4 cm

Radius of hemisphere, r = 4 cm

Height of cone, h = 6 cm

So,

$\bf \:Volume_{(solid)} = Volume_{(cone)} + Volume_{(hemisphere)}$

$\sf \: ⟼Volume_{(solid)} = \dfrac{1}{3} \pi \: {r}^{2} h + \dfrac{2}{3} \pi \: {r}^{3}$

$\sf \: ⟼Volume_{(solid)} = \dfrac{1}{3} \pi \: {r}^{2} (h + 2r)$

$\sf \: ⟼Volume_{(solid)} = \dfrac{1}{3} \times \dfrac{22}{7} \times {4}^{2} \times (6 + 2 \times 4)$

$\sf \: ⟼Volume_{(solid)} = \dfrac{1}{3} \times \dfrac{22}{7} \times 16\times 14$

$\sf \: ⟼Volume_{(solid)} = \dfrac{704}{3} \: {cm}^{3}$

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$\bf \:\large \red{AηsωeR : 2.} ✍$

☆Radius of cone, r = 4 cm

☆Radius of hemisphere, r = 4 cm

☆Height of cone, h = 6 cm

▪︎Slant height of cone, is given by

$\sf \: ⟼ {l}^{2} = {r}^{2} + {h}^{2}$

$\sf \: ⟼ {l}^{2} = {4}^{2} + {6}^{2}$

$\sf \: ⟼ {l}^{2} = 16 + 36$

$\sf \: ⟼ {l}^{2} = 52$

$\sf \: ⟼ l= \sqrt{52} = \sqrt{2 \times 2 \times 13} = 2 \sqrt{13} \: cm$

$\sf \:Surface Area_{(solid)} = Surface Area_{(cone)} + Surface Area_{(hemisphere)}$

$\sf \: ⟼Surface Area_{(solid)} = \pi \: rl + 2\pi \: {r}^{2}$

$\sf \: ⟼Surface Area_{(solid)} = \pi \: r(l + 2r)$

$\sf \: ⟼Surface Area_{(solid)} = \dfrac{22}{7} \times 4 \times (2 \sqrt{13} + 2 \times 4)$

$\sf \: ⟼Surface Area_{(solid)} = \dfrac{88}{7} \times 2 \times ( \sqrt{13} + 4)$

$\sf \: ⟼Surface Area_{(solid)} = \dfrac{176}{7} (3.605 + 4)$

$\sf \: ⟼Surface Area_{(solid)} = \dfrac{176}{7} \times 7.605$

$\sf \: ⟼Surface Area_{(solid)} = 191.21 \: {cm}^{2}$

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$\large \red{\bf \: ⟼ Explore \: more } ✍$

More 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