Question

the sum of the radius and height of a cylinder is 37cm and the total surface area of the cylinder is 1628cm2. find the volume of the cylinder.

Answer 1

Given: The sum of the radius and height of a cylinder is 37 cm. And, the Total surface area of the cylinder is 1628 cm².

Need to find: The volume of the cylinder.

❍ Let r and h be the radius and Height of the cylinder respectively.

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$\underline{\boldsymbol{According\: to \:the\: Question :}}$

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• The sum of the radius and height of a cylinder is 37 cm.

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Therefore,

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$:\implies\sf r + h = 37 \qquad\quad\bigg\lgroup\bf Equation\;(I)\bigg\rgroup$

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$\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}$⠀⠀⠀⠀

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$\star\;\boxed{\sf{\pink{TSA_{\:(cylinder)} = 2\pi r (h + r)}}}$

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where,

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• r is radius of the cylinder and h is the height of the cylinder. And the TSA(Total surface area) of the cylinder is given that is 1628 cm². Now, Comparing,

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$:\implies\sf 2 \pi r(h + r) = 1628 \\\\\\:\implies\sf 2 \pi r(37) = 1628 \\\\\\:\implies\sf 2 \times \dfrac{22}{7} \times r \times 37 = 1628\\\\\\:\implies\sf r = \dfrac{\cancel{1628}\; \times 7}{2 \times \;\cancel{22}\; \times 37} \\\\\\:\implies\sf r = \dfrac{814 \times 7}{2 \times 11 \times 37} \\\\\\:\implies\sf r = \cancel\dfrac{5698}{814}\\\\\\:\implies{\underline{\boxed{\frak{\pink{r = 7\;cm}}}}}\;\bigstar$

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$\underline{\bf{\dag} \:\mathfrak{By\;using\; Equation\;(1)\: :}}$⠀⠀⠀⠀

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$:\implies\sf r + h = 37 \\\\\\:\implies\sf h = 37 - r \\\\\\:\implies\sf h = 37 - 7 \\\\\\:\implies{\underline{\boxed{\frak{\pink{h = 30\;cm}}}}}\;\bigstar$

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$\therefore{\underline{\sf{Hence, \; radius\;and\; height\;of\; cylinder\;are\; \bf{7cm\;and\;30cm }.}}}$

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★ To calculate the Volume of cylinder formula is given by :

$\star\;\boxed{\sf{\purple{Volume_{\:(cylinder)} = \pi r^2h}}}$

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$:\implies\sf Volume_{\:(cylinder)} = \pi r^2 h\\\\\\:\implies\sf Volume_{\:(cylinder)} = \dfrac{22}{\cancel{\;7}} \times \cancel{\;7} \times 7 \times 30\\\\\\:\implies\sf Volume_{\:(cylinder)} = 22 \times 7 \times 30 \\\\\\:\implies\sf Volume_{\:(cylinder)} = 22 \times 210\\\\\\:\implies{\underline{\boxed{\frak{\purple{Volume_{\:(cylinder)} = 4620\;cm^3}}}}}\;\bigstar$

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$\therefore{\underline{\sf{Hence,\; volume\;of\; cylinder\;is\; \bf{ 4620\; cm^3}.}}}$

Answer 2

Given :-

Sum of radius and height = 37 cm

TSA = 1628 cm

To Find :-

Volume

Solution :-

$\sf 2\pi r(h + r) = 1628$

$\sf 2 \times \dfrac{22}{7} \times r (37) = 1628$

$\sf \frac{44}{7} \times r (37) = 1628$

Now

$\sf \frac{44}{7} \times r = \frac{1628}{37}$

$\sf \frac{44}{7} \times r = 44$

$\sf r = 7$

Now

r + h = 37

7 + h = 37

h = 37 - 7

h = 30

$\large \bf Volume = \pi r^{2} h$

$\sf Volume = \dfrac{22}{7} \times 7 \times 7 \times 30$

$\sf Volume = 22 \times 7 \times 30$

Volume = 4620  cm^3