A radius circular cylinder bring having diameter 12cm and height 15cm is full ice-cream.

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A cylindrical metallic pipe is 14cm long.The difference between the outside and inside surfaces is 44cm square.If the pipe is made up of 99 cubic cm of metal.Find the outer and inner radii of the pipe.

A radius circular cylinder bring having diameter 12cm and height 15cm is full ice-cream. The ice-cream is to be filled in cones of height 12cm and diameter 6cm having a hemisphere shape on top find the number of such cones which can be filled with icecream?

its solution is.....
Given:

For right circular cylinder

Diameter = 12 cm

Radius(R1) = 12/2= 6 cm & height (h1) = 15 cm

Volume of Cylindrical ice-cream container= πr1²h1= 22/7 × 6× 6× 15= 11880/7 cm³

Volume of Cylindrical ice-cream container=11880/7 cm³


For cone,

Diameter = 6 cm

Radius(r2) =6/2 = 3 cm & height (h2) = 12 cm
Radius of hemisphere = radius of cone= 3 cm

Volume of cone full of ice-cream= volume of cone + volume of hemisphere

= ⅓ πr2²h2 + ⅔ πr2³= ⅓ π ( r2²h2 + 2r2³)

= ⅓ × 22/7 (3²× 12 + 2× 3³)

= ⅓ × 22/7 ( 9 ×12 + 2 × 27)

= 22/21 ( 108 +54)

= 22/21(162)

= (22×54)/7

= 1188/7 cm³


Let n be the number of cones full of ice cream.


Volume of Cylindrical ice-cream container =n × Volume of one cone full with ice cream

11880/7 = n × 1188/7

11880 = n × 1188

n = 11880/1188= 10

n = 10

Hence, the required Number of cones = 10

Let r and R be the inner and outer radius of the cylindrical metallic pipe respectively.

h be the height of the metallic pipe = 14 cm

Difference between the Curved surface area of the outer cylinder and Curved surface area of the inner cylinder  = 2πRh - 2πrh.

Given that difference between the outside and inside curved surface area of cylinder is 44 cm² .

⇒ 2πh( R - r) = 44

⇒  44 / 7 x 14 ( R - r) = 44

⇒ R - r = 1 / 2 = 0.5 ----------(1)

Given the pipe is made up of 99 cubic cm of metal so that

Volume of cylindrical metallic pipe = πR²h - πr²h.

⇒ 22/7 x 14 (R² - r²) = 99 cm³ .

⇒ 44 x (R² - r²) = 99

⇒ (R² - r²) = 9 / 4 = 2.25

⇒ ( R - r)(R + r)  = 2.25

= (0.5)x(R + r) = 2.25

R + r = 2.25 / 0.5 = 4.5

R + r  = 4.5  ------------ (2)

Adding (1) and (2) we get

2R = 4.5 + 0.5 = 5

∴ R = 2.5 cm and r  = 2 cm

∴ Outer side radius R = 2.5 cm and inner side radius r  = 2 cm.