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Cylinder, Cone and Sphere — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsCylinder, Cone and Sphere5 Marks
Question
The difference between the outer curved surface area and the inner curved surface area of a hollow cylinder is $352 \ cm^2.$ If its height is $28 \ cm$ and the volume of material in it is $ 704 \ cm^3;$ find its external curved surface area.
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Answer

Let $R$ and $r$ be the outer and inner radii of hollow mwtallic cylinder.
Let h be the height of the metallic cylinder. It is given that
Outer curved surface area $-$ Inner curved surface area $= 352$
$\Rightarrow 2\pi Rh - 2\pi rh = 352$
$\Rightarrow 2\pi h (R - r) = 352$
$\Rightarrow 2 \times \frac{22}{7} \times 28( R - r )=352$
$\Rightarrow R - r =\frac{352 \times 7}{2 \times 22 \times 28}$
$\Rightarrow R - r = 2 ...(i)$
Volume of material in it $= 704 \ cm^3$
$\Rightarrow \pi R^2h - \pi r^2h = 704$
$\Rightarrow \pi h (R^2 - r^2) = 704$
$\Rightarrow \frac{22}{7} \times 28\left( R ^2- r ^2\right)=704$
$\Rightarrow R ^2- r ^2=\frac{704 \times 7}{22 \times 28}$
$\Rightarrow (R + r)(R-r) = 8$
$\Rightarrow (R + r) \times 2 = 8$
$\Rightarrow R + r = 4 ...(ii)$
Adding $(i)$ and $(ii)$ we get
$2R = 6 \Rightarrow R = 3 \ cm$
$\therefore$ External curved surface area $= 2\pi Rh$
$=2 \times \frac{22}{7} \times 3 \times 28=528 \ cm ^2$

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