Question

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Question : A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.​

$\large\texttt{Answer Fast :)}$

Answer 1

Required solution :

${\large{\bold{\rm{\underline{Let's \; understand \; the \; question \; 1^{st}}}}}}$

• This question says that there is a vessel and it is in the form of a hollow hemisphere mounted by a hollow cylinder. Now it's given that the diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. And at last we have to find the inner surface area of the given vessel. Let's do it..!

${\large{\bold{\rm{\underline{Given \; that}}}}}$

✯ A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. Where,

• The diameter of the hemisphere = 14 cm

• The total height of the vessel = 13 cm.

${\large{\bold{\rm{\underline{To \; find}}}}}$

★ The inner surface area of the vessel.

${\large{\bold{\rm{\underline{Solution}}}}}$

★ The inner surface area of the vessel = 571.48 cm²

${\large{\bold{\rm{\underline{Using \; concepts}}}}}$

★ Formula to find curved surface area of the cylinder.

★ Formula to find curved surface area of the hemisphere

${\large{\bold{\rm{\underline{Using \; formula}}}}}$

★ CSA of the cylinder = 2πrh

★ CSA of the hemisphere = 2πr²

${\large{\bold{\rm{\underline{Where,}}}}}$

• CSA denotes Curved Surface Area

• π is pronounced as pi

• Value of π is 22/7 or 3.14

• r dentoes radius

• h denotes height

• ² means square

${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

~ As it's already given that the hollow hemisphere mounted by a hollow cylinder. So according to this,

➙ Height of the hemispherical part = 7 cm

➙ Height of cylinder part = 13-7 = 6 cm

~ According to the question, it is also cleared that the Inner surface area of the vessel = CSA of cylinder + CSA of hemisphere.

~ Means Inner surface area of the vessel = 2πrh + 2πr²

~ Let us put the values,

➙ 2(3.14)(7)(6) + 2(3.14)(7)²

➙ 2 × 3.14 × 7 × 6 + 2 × 3.14 × (7)²

➙ 2 × 3.14 × 7 × 6 + 2 × 3.14 × 7 × 7

➙ 2 × 3.14 × 7 × 6 + 2 × 3.14 × 49

➙ 2 × 3.14 × 42 + 2 × 3.14 × 49

➙ 6.28 × 42 + 6.28 × 49

➙ 263.76 + 307.72

➙ 571.48 cm²

${\large{\bold{\rm{\underline{Additional \; knowledge}}}}}$

Cylinder diagram :

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\thicklines\multiput(-0.5,-1)(26,0){2}{\line(0,1){40}}\multiput(12.5,-1)(0,3.2){13}{\line(0,1){1.6}}\multiput(12.5,-1)(0,40){2}{\multiput(0,0)(2,0){7}{\line(1,0){1}}}\multiput(0,0)(0,40){2}{\qbezier(1,0)(12,3)(24,0)\qbezier(1,0)(-2,-1)(1,-2)\qbezier(24,0)(27,-1)(24,-2)\qbezier(1,-2)(12,-5)(24,-2)}\multiput(18,2)(0,32){2}{\sf{r}}\put(9,17.5){\sf{h}}\end{picture}$

Some formulas -

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Volume \: of \: cylinder \: = \: \pi r^{2}h}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Surface \: area \: of \: cylinder \: = \: 2 \pi rh + 2 \pi r^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Lateral \: area \: of \: cylinder \: = \: 2 \pi rh}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Base \: area \: of \: cylinder \: = \: \pi r^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Height \: of \: cylinder \: = \: \dfrac{v}{\pi r^{2}}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Radius \: of \: cylinder \: = \:\sqrt \dfrac{v}{\pi h}}}}$

Diagram of this question -

Kindly see from attachment

Answer 2

$\underline{\underline{\maltese\:\:\textbf{\textsf{Question}}}}$

• A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel

$\underline{\underline{\maltese\:\:\textbf{\textsf{Given}}}}$

• A vessel is in the form of a hollow hemisphere
• Vessel is mounted by a Hollow Cylinder
• Diameter of the hemisphere = 14 cm
• Total height of the vessel = 13 cm

$\underline{\underline{\maltese\:\:\textbf{\textsf{To Find}}}}$

• Inner Surface Area of the Vessel

$\underline{\underline{\maltese\:\:\textbf{\textsf{To Find}}}}$

• Inner Surface Area of the Vessel = 572 cm²

$\underline{\underline{\maltese\:\:\textbf{\textsf{Basic Terms}}}}$

•  Diameter : The center of a circle is the midpoint of its diameter, It divides the circle into two equal parts.  

• Radius : The distance from the center to the circumference of a circle.

$\underline{\underline{\maltese\:\:\textbf{\textsf{Formulas Used}}}}$

• Diameter = 2 × Radius
• Radius = Diameter/2
• Curved Surface Area of cylinder = 2πrh
• Curved Surface Area of hemisphere = 2πr²

$\underline{\underline{\maltese\:\:\textbf{\textsf{Calculations}}}}$

Inner Surface Area of the Vessel can be calculated by adding Curved Surface Area of Vessel and Curved Surface Area of Cylinder, In Simple Words : Inner Surface Area of the Vessel = Curved Surface Area of Hemisphere + Curved Surface Area of Cylinder

$\longrightarrow$ Inner Surface Area of the Vessel = 2πr² + 2πrh

$\longrightarrow$ Inner Surface Area of the Vessel = (2 × π × r²) + (2 × π × r × h)

• We know value of π = 22/7

$\longrightarrow$ Inner Surface Area of the Vessel = (2 × 22/7 × r²) + (2 × 22/7 × r × h)

$\longrightarrow$ Inner Surface Area of the Vessel = (44/7 × r²) + (44/7 × r × h)

• Given Diameter = 14 cm, Radius = Diameter/2 = 14/2 cm = 7 cm

$\longrightarrow$ Inner Surface Area of the Vessel = [44/7 × (7 cm)²] + (44/7 × r × h)

$\longrightarrow$ Inner Surface Area of the Vessel = [44/7 × 49 cm²] + (44/7 × r × h)

• Radius of Cylinder = Radius of Hemisphere = 7 cm

$\longrightarrow$ Inner Surface Area of the Vessel = [44/7 × 49 cm²] + [44/7 × 7 cm × h]

$\longrightarrow$ Inner Surface Area of the Vessel = [44/1 × 7 cm²] + [44/7 × 7 cm × h]

$\longrightarrow$ Inner Surface Area of the Vessel = [44 × 7 cm²] + [44/1 × 1 cm² × h]

$\longrightarrow$ Inner Surface Area of the Vessel = 308 cm² + [44 × 1 cm × h]

$\longrightarrow$ Inner Surface Area of the Vessel = 308 cm² + [44 cm × h]

• Height of Cylinder = 13 cm - 7 cm = 6 cm

$\longrightarrow$ Inner Surface Area of the Vessel = 308 cm² + [44 cm × 6 cm]

$\longrightarrow$ Inner Surface Area of the Vessel = 308 cm² + 264 cm²

$\longrightarrow$ Inner Surface Area of the Vessel = 572 cm²