Question

$\large{ \red{ \tt{Question࿐}}}$
If the height of cone is doubled then its volume increased by
a) 100%
b) 200%
c) 150%
d) 10.0%

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Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

Radius of cone be r units.

Height of cone be h units

So, Volume of cone of height h and radius r is given by

$\color{green}\boxed{ \rm{ \:V_1 \: = \: \frac{1}{3} \: \pi \: {r}^{2} \: h \: \: }}$

Now,

Radius of cone be r units.

Height of cone be 2h units.

So, Volume of cone of radius r and height 2h is given by

$\rm \:V_2 \: = \: \frac{1}{3} \: \pi \: {r}^{2} \: \times 2 \: h \: \:$

So,

$\color{green}\boxed{ \rm{ \:V_2 \: = \: \frac{2}{3} \: \pi \: {r}^{2} \: h \: \: }}$

Now,

$\rm \: \% \: age \: increase \: in \: volume \: is \:$

$\rm \: = \: \dfrac{V_2 - V_1}{V_1} \times 100 \: \%$

$\rm \: = \: \dfrac{\dfrac{2}{3} \pi \: {r}^{2} h \: - \: \dfrac{1}{3} \pi \: {r}^{2} h}{\dfrac{1}{3} \pi \: {r}^{2} h} \times 100\% \:$

$\rm \: = \: \dfrac{\dfrac{1}{3} \pi \: {r}^{2} h \: }{\dfrac{1}{3} \pi \: {r}^{2} h} \times 100\% \:$

$\rm \: = \: 100 \: \%$

Hence,

$\rm\implies \:\rm \: \% \: age \: increase \: in \: volume \: is \: = \: 100 \: \%$

So, option (a) is correct.

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Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{CSA_{(cylinder)} = 2\pi \: rh}\\ \\ \bigstar \: \bf{Volume_{(cylinder)} = \pi {r}^{2} h}\\ \\ \bigstar \: \bf{TSA_{(cylinder)} = 2\pi \: r(r + h)}\\ \\ \bigstar \: \bf{CSA_{(cone)} = \pi \: r \: l}\\ \\ \bigstar \: \bf{TSA_{(cone)} = \pi \: r \: (l + r)}\\ \\ \bigstar \: \bf{Volume_{(sphere)} = \dfrac{4}{3}\pi {r}^{3} }\\ \\ \bigstar \: \bf{Volume_{(cube)} = {(side)}^{3} }\\ \\ \bigstar \: \bf{CSA_{(cube)} = 4 {(side)}^{2} }\\ \\ \bigstar \: \bf{TSA_{(cube)} = 6 {(side)}^{2} }\\ \\ \bigstar \: \bf{Volume_{(cuboid)} = lbh}\\ \\ \bigstar \: \bf{CSA_{(cuboid)} = 2(l + b)h}\\ \\ \bigstar \: \bf{TSA_{(cuboid)} = 2(lb +bh+hl )}\\ \: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$