Question

From a solid cylinder whose height is 8 cm and radius 6 cm, a conical
cavity of height 8 cm and of base radius 6 cm, is hollowed out. Find the
volume of the remaining solid. Also, find the total surface area of the
remaining solid. Take a = = 3.14.
(CBSE 2009

Answer 1

Answer:

Height of cylinder = 8 cm

Radius of cylinder = 6 cm

Volume of cylinder = πR²H

=> 22/7 × 6 × 6 × 8

=> 22 × 6 × 6 × 8/7

=>6336/7

=> 905.14 cm²

Height of Cone = 8 cm

Radius of cone = 6 cm

Slant Height (L) = ✓(H)²+(R)² => ✓(8)²+(6)²

Slant Height (L) = ✓64+36 = ✓100 = 10 cm

Volume of cone = 1/3πR²H

=> 1/3 × 22/7 × 6 × 6 × 8

=> 22 × 6 × 6 × 8/21

=> 6336/21

=>301.71 cm².

Volume of remaining solid = Volume of cylinder- Volume of cone

=> πR²H - 1/3πR²H

=> 2/3 πR²H

=> 2/3 × 22/7 × 6 × 6 × 8

=> 2 × 22 × 6 × 6 × 8/21

=> 12672/21

=> 603.42 cm².

TSA of remaining solid = CSA of cylinder +CSA of cone + Area of upper circular face of cone

=> 2πRH + πRL + πR²

=> πR(2H+L+R)

=> 22/7 × 6 ( 2 × 8 + 10 + 6)

=> 132/7 ( 16 + 16)

=> 132/7 × 32

=> 132 × 32/7

=> 4224/7

=> 603.42 cm².

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Answer 2

$\huge\mathfrak{Answer:}$

Given:

• We have been given that the height of the solid cylinder is 8cm and radius of its base is 6cm.
• Height of the conical cavity = 8cm and radius of its base = 6cm.

To Find:

• We need to find the area of the remaining solid and we also need to find the total surface area of the remaining solid.

Solution:

As it is given that the height of cylinder = 8cm and radius of its base is 6cm.

We know that the volume of cylinder is : πr²h

Substituting the values, we have

$\implies\sf{3.14 \times 6 \times 6 \times 8}$

$\implies\sf{3.14 \times 36 \times 8}$

$\implies\sf{3.14 \times 288}$

$\implies\sf{904.32 {cm}^{2}}$

Now, it is also given that the height of the conical cavity = 8cm and radius of its base is 6cm.

We know that,

Slant Height (l) = ✓{(h)²+(r)²}

$\sf l = \sqrt{8^2 + 6^2}$

$\implies \sf l = \sqrt{64 + 36}$

$\implies\sf{l = \sqrt{100}}$

$\implies\sf{10cm}$

Now, we know that,

$\sf{volume \: of \: cone \: is \: \dfrac{1}{3}\pi {r}^{2} h}$

Substituting the values, we have

$\implies\sf{ \dfrac{1}{3} \times 3.14 \times 6 \times 6 \times 8}$

$\implies\sf{3.14\times2 \times 6 \times 8}$

$\implies\sf{301.44 {cm}^{2} }$

Now, Volume of remaining solid = Volume of cylinder- Volume of cone. We have

$\sf{904.32 - 301.44}$

$\sf{602.88}$

Now, TSA of remaining solid = CSA of cylinder + CSA of cone + Area of upper circular face of cone.

$\implies\sf{2\pi r h + \pi r l + \pi {r}^{2} }$

$\implies\sf{\pi r(2h + l + r)}$

Substituting the values, we have

$\implies\sf{3.14 \times 6(2 \times 8 + 10 + 6)}$

$\implies\sf{18.84(16 + 16)}$

$\implies\sf{18.84 \times 32}$

$\implies\sf{ 602.88 {cm}^{2} }$

Hence, the total surface area of the remaining solid is 602.88cm².