Question

radius of a right circular cylinder is increased by 15 percent and it's height decreased by 20 percent then by what percent sell the volume of the cylinder change

Answer 1

Appropriate Question :-

If the radius of a right circular cylinder is increased by 15 percent and it's height is decreased by 20 percent, then by what percent the volume of the cylinder change?

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

Let assume that

• Radius of cylinder = r units

• Height of cylinder = h units

We know,

Volume of cylinder of radius r and height h is given by

$\rm \: Volume_{(Cylinder)} = \pi \: {r}^{2} h \: \cdots \cdots(1)$

Now, According to statement

Radius is increased by 15 % and height is decreased by 20%.

So, New dimensions are

$\rm \: R = r + 15\% \: of \: r$

$\rm \: = r + \frac{15}{100} \: \times \: r$

$\rm \: = r + \frac{3r}{20}$

$\rm \: = \frac{20r + 3r}{20}$

$\rm \: = \frac{23r}{20}$

Now,

$\rm \: H = h - 20\% \: of \: h$

$\rm \: H = h - \frac{20}{100} \times h$

$\rm \: H = h - \frac{h}{5}$

$\rm \: H = \frac{5h - h}{5}$

$\rm \: H = \frac{4h}{5}$

Now, Volume of cylinder is given by

$\rm \: Volume_{(Cylinder)}' = \pi \: {R}^{2} H$

$\rm \: = \: \pi \: {\bigg[\dfrac{23r}{20} \bigg]}^{2} \times \dfrac{4h}{5}$

$\rm \: = \: \pi \: \times \dfrac{529 {r}^{2} }{400} \times \dfrac{4h}{5}$

$\rm \: = \:\dfrac{529}{500} \pi \: {r}^{2} h$

$\rm\implies \:Volume_{(Cylinder)}' = \:\dfrac{529}{500} \pi \: {r}^{2} h$

Now,

$\rm \: \% \: age \: change \: in \: Volume_{(Cylinder)}$

$\rm \: = \: \dfrac{Volume_{(Cylinder)}' - Volume_{(Cylinder)}}{Volume_{(Cylinder)}} \times 100\%$

$\rm \: = \: \dfrac{\dfrac{529}{500}\pi \: {r}^{2} h - \pi \: {r}^{2} h }{\pi \: {r}^{2} h} \times 100\%$

$\rm \: = \: \dfrac{\bigg(\dfrac{529}{500}- 1\bigg)\pi \: {r}^{2} h }{\pi \: {r}^{2} h} \times 100\%$

$\rm \: = \: \bigg(\dfrac{529 - 500}{500} \bigg) \times 100\%$

$\rm \: = \: 5.8\: \% \:$

Hence,

$\red{\rm\implies \% \: age \: change \: in \: Volume_{(Cylinder)} = 5.8\%} \:$

$\rule{190pt}{2pt}$

Additional Information :-

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