Question

The numerical value of the volume and whole surface area of a solid right circular cylinder are equal. If its height is h and radius r then express h in terms of r.​

Answer 1

Basic Formula's Used :-

We know that,

$\green{\boxed{ \bf \:Volume_{(Cylinder)} = \pi \: {r}^{2}h}}$

$\blue{\boxed{ \bf \:TSA_{(Cylinder)} = 2\pi \: r(h + r)}}$

where,

• r = radius of cylinder

• h = height of cone

• TSA = Total Surface Area of cylinder

Let's solve the problem now!!!

$\green{\large\underline{\bf{Solution-}}}$

Let

• Radius of cylinder = r units

• Height of cylinder = h units

According to statement,

It is given that,

Volume of Cylinder = Total Surface Area of Cylinder

$\red{\bf :\longmapsto\:Volume_{(Cylinder)} = TSA_{(Cylinder)}}$

$\rm :\longmapsto\:\pi \: {r}^{2}h = 2\pi \: r(h + r)$

$\rm :\longmapsto\:rh = 2(r + h)$

$\rm :\longmapsto\:rh = 2r +2h$

$\rm :\longmapsto\:rh - 2h= 2r$

$\rm :\longmapsto\:h(r - 2) = 2r$

$\pink{\bf\implies \:h = \dfrac{2r}{r - 2}}$

Additional Information :-

• Volume of cylinder = πr²h

• T.S.A of cylinder = 2πrh + 2πr²

• Volume of cone = ⅓ πr²h

• C.S.A of cone = πrl

• T.S.A of cone = πrl + πr²

• Volume of cuboid = l × b × h

• C.S.A of cuboid = 2(l + b)h

• T.S.A of cuboid = 2(lb + bh + lh)

• C.S.A of cube = 4a²

• T.S.A of cube = 6a²

• Volume of cube = a³

• Volume of sphere = 4/3πr³

• Surface area of sphere = 4πr²

• Volume of hemisphere = ⅔ πr³

• C.S.A of hemisphere = 2πr²

• T.S.A of hemisphere = 3πr²

Answer 2

Refer to the attachment