A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream
Solution:
A figure is drawn below to visualize the shapes according to the given question.
From the above figure, it can be seen that the diameter of the hemisphere is the same as the cone.
As the ice cream is to be filled into cones having a hemispherical shape on the top, the volume of ice cream filled into the cone includes the volume of the cone and volume of the hemisphere.
The volume of the ice cream in each cone = volume of the cone + volume of the hemisphere
Since the ice cream is to be filled from a cylindrical container, the volume of the ice cream filled into cones will be the same as the volume of the ice cream in the cylindrical container.
Volume of the ice cream in cylindrical container = number of cones filled with ice cream × volume of the ice cream in each cone
Hence, number of cones filled with ice cream = volume of the ice cream in cylindrical container ÷ volume of the ice cream in each cone
Let us find the number of the ice cream cones filled by using the formulae:
Volume of the hemisphere = 2/3 πr3, where r is the radius of the hemisphere
Volume of the cone = 1/3 πr2h, where r and h are the radius and height of the cone respectively
Volume of the cylinder = πr2h where r and h are radius and height of the cylinder respectively.
Let the height of cylindrical container be H = 15 m
Radius of cylindrical container, R = d/2 = 12/2 m = 6 m
Radius of the ice-cream cone = Radius of the hemispherical top, r = 6/2 m = 3 m
Height of the ice-cream cone, h = 12 m
Let 'n' ice-cream cones be filled with ice-cream from the cylindrical container.
Volume of the ice cream in container = n × volume of the ice cream in each cone
πR2H = n × (1/3 πr2h + 2/3 πr3)
R2H = (1/3) nr2(h + 2r)
n = 3R2H / r2(h + 2r)
= [3 × (6 cm)2 × 15 cm] / [(3cm)2 × (12 cm + 2 × 3 cm)]
= (3 × 36 cm2 × 15 cm) / (9 cm2 × 18 cm)
= 10
Therefore, 10 ice cream cones can be filled with the ice cream in the container.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 13
Video Solution:
A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream
NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.3 Question 5
Summary:
If a container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream and the ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top then the number of such cones which can be filled with ice cream is 10.
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