Question

The area of the curved surface area of a cylinder is three times the base area and height of the cylinder exceeds the radius by 7cm .Find the volume of the cylinder

Answer 1

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

Given that, height of cylinder exceeds the radius by 7 cm

Let assume that

Radius of cylinder = r cm

Height of cylinder = h = r + 7

According to statement, it is further given that the curved surface area of a cylinder is three times the base area.

We know,

Curved Surface Area of cylinder of radius r and height h is given by

$\boxed{ \rm{ \:CSA_{(Cylinder)} \: = \: 2 \: \pi \: r \: h \: \: }}$

and

$\boxed{ \rm{ \:Base \: Area_{(Cylinder)} \: = \: \pi \: {r}^{2} \: \: }}$

So,

$\rm \: 2\pi \: rh \: = \: 3\pi {r}^{2}$

$\rm \: 2h \: = \: 3r$

On substituting the value of h, we get

$\rm \: 2 \:(r + 7) \: = \: 3r$

$\rm \: 2r + 14 \: = \: 3r$

$\rm \: 3r - 2r = 14$

$\rm\implies \:r \: = \: 14 \: cm$

So, we have

Radius of cylinder, r = 14 cm

Height of cylinder, h = r + 7 = 14 + 7 = 21 cm

Now,

$\rm \: Volume _{(Cylinder)} \: = \: \pi \: {r}^{2} \: h$

$\rm \: =  \:\dfrac{22}{7} \times 14 \times 14 \times 21$

$\rm \: =  \:22 \times 2 \times 14 \times 21$

$\rm \: =  \:12936 \: {cm}^{3}$

Hence,

$\color{green}\rm\implies \:\boxed{ \rm{ \:Volume _{(Cylinder)} = 12936 \: {cm}^{3} \: }}$

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

Additional Information

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

Answer 2

Given:

The height of the cylinder exceeds the radius by 7cm.

The CSA of the cylinder is 3 times the base area.

To Find:

The volume of the cylinder

Solution:

Let the radius of the cylinder be r

Height of the cylinder h = r + 7

It is given that the curved surface area of the cylinder is three times the base area.

The formula to calculate the curved surface area of the cylinder is

CSA of the cylinder = 2πrh

The base area of the cylinder can be calculated by using πr²

Now, substituting the values

2πrh = 3πr²   [ it is given that 3 times the base area]

2h = 3r

Then we put the value of h we get

⇒ 2(r+7) = 3r

⇒ 2r + 14 = 3r

⇒ 3r - 2r = 14

⇒ r = 14cm

∴ The radius of the cylinder is 14cm.

The height of the cylinder h = r + 7

                                            h = 14 + 7

                                             h = 21cm

∴ The height of the cylinder = 21cm

Now, Volume of the cylinder = πr²h

                                                = 22/7 × 14 × 14 × 21

                                                = 22 × 2 × 14 × 21

                                                = 12936cm³

Therefore the volume of the cylinder = 12936cm³.