Ex.13.7 Q2 Surface Areas and Volumes Solution - NCERT Maths Class 9


Question

Find the capacity in litres of a conical vessel with

(i) radius \(7\rm\, cm,\) slant height \(25\rm\, cm\)

(ii) height \(12\rm\, cm,\) slant height \(13\rm\, cm\)

 Video Solution
Surface-Areas-And-Volumes
Ex exercise-13-7 | Question 2

Text Solution

Reasoning:

Capacity of a conical vessel is nothing but the Volume of the cone \(\begin{align} = \frac{1}{3}\pi {r^2}h \end{align}\)

What is  known?

Radius \((r) = 7\rm\, cm\)

Slant height \((1) = 25\rm\, cm\)

What is  unknown?

Capacity of the vessel in litres.

Steps:

Capacity of the conical vessel \(=\) volume of cone \(\begin{align} = \frac{1}{3}\pi {r^2}h \end{align}\)

Where \(\begin{align}h = \sqrt {{l^2} - {r^2}} \end{align}\)

Radius \(\begin{align}(r) = 7\rm \,\rm\,cm \end{align}\)

Slant height \(\begin{align}(l) = 25\rm\,\rm \,cm \end{align}\)

\[\begin{align}\therefore h &= \sqrt {{{(25)}^2} - {7^2}} \\ &= \sqrt {(25 + 7)(25 - 7)}\\ {\rm{ [ (}}{{\rm{a}}^2} - {b^2}) &= (a + b)(a - b)]\\ &= \sqrt {32 \times 18} \\ &=\! \sqrt {2 \!\times\! 2\! \times\! 2\! \times \!2\! \times \!2 \!\times\! 3 \!\times \!3\! \times\! 2} \\ &= \sqrt {576} \\ &= 24\,\,\rm\,cm \end{align}\]

Volume
\[\begin{align}& = \frac{1}{3} \times \frac{{22}}{7} \times 7 \times 7 \times 24\\ &= 1232\,\,\rm\,c{m^3}\,\,or\,\,1.232\,\text{l} \end{align}\]

\[\begin{align} [\therefore 1000\,\,c{m^3} = 1\,l] \end{align}\]

What is  known?

Radius \((r) = 12 \rm\,cm\)

Slant height \(\begin{align}(l) = 13\,\,\rm\,cm \end{align}\)

What is  unknown?

Capacity of the vessel in litres.

Steps:

Capacity of the conical vessel \(=\) volume of cone \(\begin{align} = \frac{1}{3}\pi {r^2}h \end{align}\)

Height \((h) = 12\rm\, cm\)

Slant height \(\begin{align}(l) = 13\,\,\rm\,cm \end{align}\)

\[\begin{align}&{r^2} + {h^2} = {l^2} \\ &{r^2} = {l^2} - {h^2} \\ &r=\sqrt{l^{2}-h^{2}} \\ &= \sqrt{(13)^{2}-(12)^{2}} \\ &=\sqrt{(13+12)(13-12)} \\&= \sqrt {25} \\ &= 5\,\,\rm\,cm \end{align}\]

Capacity

\[\begin{align}&= \frac{1}{3} \times \frac{{22}}{7} \times 5 \times 5 \times 12\\&= \frac{{2200}}{7}\\& = \frac{2200}{7000}\\& = \frac{11}{35}l \end{align}\]

Answer:

Capacity \(=1.232\,\, l\)

Capacity \(\begin{align}=\frac{{11}}{{35}} \,l \end{align}\)

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