Question

A cylindrical metallic pipe is 14 cm long. The difference between the outside and inside surfaces is 44 cm². If the pupe is made up of 99 cu cm of metal,find the outer and inner radii of pipe.​

Answer 1

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$\huge\underline\bold\orange{Question}$

A cylindrical metallic pipe is 14 cm long. The difference between the outside and inside surfaces is 44 cm². If the pupe is made up of 99 cu cm of metal,find the outer and inner radii of pipe.

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$\huge\underline\bold\orange{Solution}$

Let tje outer and inner radii be R cm and r cm respectively.

Also, it is given that h = 14 cm. Then,

$outside \: surface \: area = (2\pi \: Rh)sq \: units \\ = (2 \times \frac{22}{7} \times R \times 14) {cm}^{2} \\ = (88R) {cm}^{2}$

$inside \: surface \: area = (2\pi \: rh)sq \: units \\ = (2 \times \frac{22}{7} \times r \times 14) {cm}^{2} = (88r) {cm}^{2} .$

$\therefore \: (88R - 88r) = 44 = > 88(R - r) = 44 = > (R - r) = \frac{1}{2} \: \: \: \: \: ....(i)$

$external \: volume \: = (\pi \: {R}^{2} h) \\ = ( \frac{22}{7} \times {R}^{2} \times 14) {cm}^{2} = ( {44R}^{2} ) {cm}^{3} .$

$internal \: volume = (\pi \: {r}^{2} h)cubic \: units \\ = ( \frac{22}{7} \times {r}^{2} \times 14) {cm}^{3} = ( {44r}^{2} ) {cm}^{3} .$

$volume \: of \: metal = (external \: volum) - (internal \: volume) \\ = ( {44R}^{2} - {44r}^{2}) {cm}^{3} = 44( {R}^{2} - {r}^{2} ) {cm}^{3} .$

$\therefore \: 44( {R}^{2} - {r}^{2} ) = 99 = > ( {R}^{2} - {r}^{2} ) = \frac{99}{44} = > ( {R}^{2} - {r}^{2} ) = \frac{9}{4} \: \: ...(ii)$

$on \: diving \: (ii) \: by \: (i) \: we \: get \: r = \frac{5}{2} \: and \: r = 2. \: \: \: \: \: \: ....(iii)$

$on \: solving \: (i) \: and \: (iii) \: we \: get \: R = \frac{5}{2} and \: r = 2.$

Hence, the outer radius = 2.5 cm and the inner radius = 2 cm.

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Answer 2

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