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Mathematics

The total surface area of a solid cylinder is 616 cm2. If the ratio between its curved surface area and total surface area is 1 : 2; find the volume of the cylinder.

Mensuration

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Answer

Given,

Total surface area = 616 cm2

⇒ 2πr(h + r) = 616

⇒ πr(h + r) = 6162\dfrac{616}{2}

⇒ πr(h + r) = 308 ……….(1)

Ratio between its curved surface area and total surface area = 1 : 2

Curved surface areaTotal surface area=122πrh2πr(h+r)=12hh+r=122h=h+r2hh=rh=r.\Rightarrow \dfrac{\text{Curved surface area}}{\text{Total surface area}} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{2πrh}{2πr(h + r)} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{h}{h + r} = \dfrac{1}{2} \\[1em] \Rightarrow 2h = h + r \\[1em] \Rightarrow 2h - h = r \\[1em] \Rightarrow h = r.

Substituting value of h in equation (1), we get :

⇒ πr(r + r) = 308

⇒ πr.2r = 308

⇒ 2πr2 = 308

⇒ πr2 = 154

227×r2\dfrac{22}{7} \times r^2 = 154

⇒ r2 = 154×722\dfrac{154 \times 7}{22}

⇒ r2 = 49

⇒ r = 49\sqrt{49}

⇒ r = 7 cm

⇒ h = 7 cm.

Volume of cylinder = πr2h

= 227×(7)2×7\dfrac{22}{7} \times (7)^2 \times 7

= 22 × 49

= 1078 cm3.

Hence, volume of cylinder = 1078 cm3.

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