A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is $\frac{8}{9}$ of the curved surface of wholecone, find the ratio of the line-segment into which the cone's altitude is divided by the plane

Question

A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is $\frac{8}{9}$ of the curved surface of wholecone, find the ratio of the line-segment into which the cone's altitude is divided by the plane

Answer 1

$l_{1}, h_{1}, r_{1}$ are length,height,radius if a short cone
$\frac{l _{1}}{l} = \frac{h _{1}}{h} = \frac{r _{1}}{r}$
$∴$ C.S.A of the smaller cone $= \frac{1}{9} \times$ C.S.A of given cone
$∵$ C.S.A. of the remaining portion $= \frac{8}{9} \times$ C.S.A. of given cone
$\Rightarrow π r_{1} l_{1} = \frac{1}{9} π r l \Rightarrow \frac{r _{1} l _{1}}{r l} = \frac{1}{9}$
or $\frac{h _{1}}{h} \times \frac{h _{1}}{h} = \frac{1}{9}$
$\Rightarrow (\frac{h _{1}}{h})^{2} = \frac{1}{9}$ or $\frac{h _{1}}{h} = \frac{1}{3}$
$h_{1} = \frac{1}{3} h ∴ h_{2} = h - h_{1} = h - \frac{1}{3} h = \frac{2}{3} h$
Hence, $\frac{h _{1}}{h _{2}} = \frac{1}{2}$ .

Answer 2

l_{1}, h_{1}, r_{1}

are length,height,radius if a short cone

\frac{l _{1}}{l} = \frac{h _{1}}{h} = \frac{r _{1}}{r}

∴

C.S.A of the smaller cone

= \frac{1}{9} \times

C.S.A of given cone

∵

C.S.A. of the remaining portion

= \frac{8}{9} \times

C.S.A. of given cone

\Rightarrow π r_{1} l_{1} = \frac{1}{9} π r l \Rightarrow \frac{r _{1} l _{1}}{r l} = \frac{1}{9}

or

\frac{h _{1}}{h} \times \frac{h _{1}}{h} = \frac{1}{9}

\Rightarrow (\frac{h _{1}}{h})^{2} = \frac{1}{9}

or

\frac{h _{1}}{h} = \frac{1}{3}

h_{1} = \frac{1}{3} h ∴ h_{2} = h - h_{1} = h - \frac{1}{3} h = \frac{2}{3} h

Hence,

\frac{h _{1}}{h _{2}} = \frac{1}{2}

.

A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is $\frac{8}{9}$ of the curved surface of wholecone, find the ratio of the line-segment into which the cone's altitude is divided by the plane

$\frac{1}{2}$

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{1}{5}$

Answer

Practice important Questions

Foundation Class 10 Maths Chapter 13

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A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is 98​ of the curved surface of wholecone, find the ratio of the line-segment into which the cone's altitude is divided by the plane

21​

31​

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Answer

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Take Toppr Scholastic Test for Aptitude and Reasoning

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A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is $\frac{8}{9}$ of the curved surface of wholecone, find the ratio of the line-segment into which the cone's altitude is divided by the plane

$\frac{1}{2}$

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{1}{5}$

If aabb is a four digit number and also a perfect square then the value of $a + b$ is

If x & y are in direct proportion and if $x = 20$ at proportionality constant $= 4$ , find y.

If $α$ and $β$ are the zeroes of polynomial $f (x) = x^{2} - p (x + 1) + c$ such that $(α + 1) (β + 1) = 0$ , then find the value of c.

Solve the equation $2 x - 6 + x + 4 = 5$

If $tan θ = \frac{4}{3}$ , then evaluate $\frac{3 sin θ + 2 cos θ}{3 sin θ - 2 cos θ}$

If $x = p - 2$ and $\frac{x}{p} - \frac{x + 1}{p} = 1$ , find $p$ .

Solve:
$\frac{3}{x} - \frac{2}{y} = 0$
$\frac{2}{x} + \frac{5}{y} = 19$
Hence, find $^{'} a^{'}$ if $y = a x + 3$

State True or False
$(x^{2} + a^{2})^{7} = (x^{7} - 21 x^{5} a^{2} + 35 x^{3} a^{4} - 7 x a^{6})^{2} + (7 x^{6} a - 35 x^{4} a^{3} + 21 x^{2} a^{5} - a^{7})^{2}$