Question

If the ratio of curved surface area and total
surface area of a cone is 3:4, then the ratio of
height and radius of the cone is
(1)4√2:1
(2)2√2:1
(3)3:1
(4)21​

Answer 1

Given that,

• The ratio of curved surface area and total surface area of a cone is 3:4

☯ Let the Height and Radius of the cone be r and h.

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Therefore,

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$:\implies\sf CSA_{cone} : TSA_{cone} = 3:4\\\\\\:\implies\sf \dfrac{\cancel{\pi \: r}l}{\cancel{\pi \: r} (l + r)} = \dfrac{3}{4} \\\\\\:\implies\sf \dfrac{l}{l + r} = \dfrac{3}{4}\\\\\\:\implies\sf 4l = 3l + 3r\\\\\\:\implies\sf l = 3r\qquad\bigg\lgroup\bf Squaring \ Both \ sides \: \bigg\rgroup\\\\\\:\implies\sf l^2 = 9r^2\\\\\\:\implies\sf r^2 + h^2 = 9 r^2 \\\\\\:\implies\sf 8r^2 = h^2\\\\\\:\implies\sf \Big(2\sqrt{2}\Big) r^2 = h^2\\\\\\:\implies\sf \bigg(\dfrac{r}{h}\bigg) = \bigg(\dfrac{2\sqrt{2}}{1}\bigg)\\\\\\:\implies{\underline{\boxed{\frak{\pink{r : h = 2\sqrt{2}: 1}}}}}\:\bigstar$

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$\therefore\:{\underline{\sf{Hence, \ the \ ratio \ of \ height \ and \ radius \ of \ cone \ is \ \bf{Option \ b) \: 2\sqrt{2} :1}.}}}$

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$\qquad\qquad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar\: Formulae\:related\:to\:cone :}}}}}\mid}$

• $\sf Area\:of\:base = \bf{\pi r^2}$

• $\sf Curved\:surface\:area\:of\:cone = \bf{\pi rl}$

• $\sf Total\:surface\:area\:of\:cone = \bf{\pi r(r + l)}$

• $\sf Volume\:of\:cone = \bf{\dfrac{1}{3} \pi r^2 h}$

Answer 2

Answer:

Given :-

The ratio of curved surface area and total surface area of a cone is 3:4

To Find :-

Ratio of height and Radius

Solution :-

As we know that

CSA = πrl

TSA = πr(l + r)

CSA/TSA = 3/4

πrl/πr(l + r) = 3/4

l/(l + r) = 3/4

• Cross multiplication

4(l) = 3(l + r)

4l = 3l + 3r

4l - 3l = 3r

l = 3r

• By squaring

l² = 9r²

r² + h² = 9r²

9r² - r² = h²

8r² = h²

r/h = 2√2:1

Hence :-

$\large \sf \:Required \: Ratio \: = 2 \sqrt{2} \ratio \: 1$