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Surface Areas and Volumes

Short Answer Type

741.

The volume of a hemisphere is 2425 $\frac{1}{2}$ cm³^. Find its curved surface area.

$[Use π = \frac{22}{7}]$

Long Answer Type

742.
From a solid cylinder of height 7 cm and base diameter 12 cm, a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid. $[Use π = \frac{22}{7}]$

OR

A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, then find the radius and slant height of the heap.

Given: Radius of cylinder = radius of cone = r = 6 cm.

Height of the cylinder = height of the cone = h = 7 cm.

Slant height of the cone = l cm.

$\sqrt{7^{2} + 6^{2}} = \sqrt{85} cm .$

Total surface area of the remaining solid = curved surface area of the

cylinder + area of the base of the cylinder + curved surface area of the cone.

$= (2 πrh + {πr}^{2} + πrl) = 2 x \frac{22}{7} x 6 x 7 + \frac{22}{7} x 6 x 6 + \frac{22}{7} x 6 x \sqrt{85} = 264 + \frac{792}{7} + \frac{132}{7} \sqrt{85} = 264 + 113.14 + \frac{132}{7} \sqrt{85} = 377.14 + \frac{132}{7} \sqrt{85} {cm}^{2}$

Volume if the conical heap = volume of the sand emptied from the bucket.

Volume of the conical heap = $\frac{1}{3} {πr}^{2} h$

= $\frac{1}{3} {πr}^{2} x 24 {cm}^{2}$ .....(Height of the conies 24cm)

...........(1)

Volume of the sand in the bucket = $= {πr}^{2} h = π x {(18)}^{2} x 32 {cm}^{2} . . . . . . . . . (2)$

Equating (1) and (2)

$= \frac{1}{3} {πr}^{2} x 24 = π x (18)^{2} x 32 \Rightarrow r^{2} = \frac{(18)^{2} x 32 x 3}{24} \Rightarrow r^{2} = \frac{18 x 18 x 32 x 3}{24} \Rightarrow r^{2} = 1296 \Rightarrow r = 36 cm$

743.

A bucket is in the form of a frustum of a cone and its can hold 28.49 litres of water. If the radii of its circular ends are 28 cm and 21 cm, find the height of the bucket. $[use π = \frac{22}{7}]$

744.

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 7 cm and the height of the cone is equal to its diameter. Find the volume of the solid.

$[Use π = \frac{22}{7}]$