10.2 Surface Area of the combination of Solids

10.2 Surface Area of the combination of Solids Solids which are made up of combination of solids in our daily life Wooden things

10.2 Surface Area of the combination of Solids House Items

10.2 Surface Area of the combination of Solids Medicine Capsules

10.2 Surface Area of the combination of Solids Bottles

10.2 Surface Area of the combination of Solids Oil Tankers

10.2 Surface Area of the combination of Solids Water tankers

10.2 Surface Area of the combination of Solids Ice cream cone shapes

10.2 Surface Area of the combination of Solids If a Oil tanker is observed carefully, it is made up of a cylinder with two hemisphere at its ends. Total Surface area of a Oil Tanker or Water tanker = Curved Surface area of a one hemisphere Curved Surface area of cylinder Curved Surface area of other hemisphere

10.2 Surface Area of the combination of Solids Devarsha wants to make a toy by putting together a hemisphere and cone . Let us see the steps that he should be going through. Step - 1 : First he should take a cone and hemisphere and bring their flat faces toghther Step – 2 : He should take the base radius of the cone equal to the radius of the hemisphere. Step – 3 : He should add the cone and hemisphere Total Surface area of the toy = Curved Surface area of the Cone Curved Surface area of the hemisphere

Try this Use known solid shape and make as many objects ( by combining more than two ) as possible that you come across in your daily life. Hint:Use clay or balls,pipes,Paper cones,boxes like cube , cuboids etc.

Try this

Think - Discuss A sphere is inscribed in cylinder . Is the surface of the sphere equal to the curved surface of the cylinder ? If yes , explain how ? Solution : Yes , the surface of the sphere equal to the curved surface of the cylinder Let the radius of cylinder be r and height be h A sphere is inscribed in cylinder . So the radius of the sphere is equal to cylinder Radius of the sphere = r Let Hight of the cylinder be h . Curved surface area of the cylinder Surface area of the sphere The surface of the sphere = the curved surface of the cylinder

Example – 8 : A right triangle , whose base and height are 15 cm and 20 cm respectively is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed . Solution : LetABC be a right triangle 20 cm 15 cm Given that base AB= 15 cm and height AC= 20 cm 25 cm Using Pythagoras theorem in ∆ ABC 20 cm 15 cm cm Let OA= x cm and OB= y cm In and We have and So by angle – angle criterion of similarity , we have ∆ BOA ~∆ BAC and and

OA= x = 12 cm and OB = y = 9 cm When ABC is revolved about the hypotenuse, we get a double cone as shown in the figure. Volume of the double cone = Volume of the cone Volume of the cone 20 cm 15 cm 25 cm 20 cm 15 cm

Surface area of the double cone = Curved Surface area of the cone Curved Surface area of the cone 20 cm 15 cm 25 cm 20 cm 15 cm

Example - 9 A wooden toy rocket is in the shape of a cone mounted on a cylinder as shown in the adjecent figure. The height of the entire rocket is 26 cm. while the height of the conical part is 6cm. The base of the conical position has a diameter of 5 cm, while diameter of the base diameter of the cylinderical portion is 3 cm. If the conical portion is to be painted orange and cylinderical portion is to be painted yellow . Find the area of the rocket painted with each of these colour 6.5 cm 6.5 cm 6 cm 2.5 cm 26 cm 5òÜ….Ò$. Solution : The height of a wooden rocket h = 26 cm The height of a Conical Part h = 6 cm diameter a Conical Part = 5 cm 1.5 cm radius of the base of the cone = 3 cm Slant height of the cone Base diameter a Cylinderical Portion d = 3 cm Height of a Cylinderical Portion h1= height of the wooden toy – height of the conical Part = 26 – 6 = 20 cm

Area of the conical portion to be painted orange colour = Curved surface area of the cone = 6 cm 6.5cm Area to be painted Yellow = Curved surface area of the cylinder + Area of the base of the cylinder = 26 cm 2.5cm 5òÜ….Ò$. 1.5cm 3 cm Area to be painted yellow

Exercise - 10.2 1. A toy is in the form of a cone mounted on a hemisphere. The diameter of the base and the height of the cone are 6 cm and 4 cm respectively . Determine the surface area of the toy. Solution : diameter of the base of the conical toy = 6 cm 5 cm Radius of conical toy = 4 cm Height of the cone Slant height of the cone 6 cm Surface area of the toy = Curved surface area of the Cone + Curved surface area of the hemisphere = Surface area of the toy

Exercise - 10.2 2. A Solid is in the form of a right circular cylinder with a hemisphere at one end and a cone at the other end. The radius of the common base is 8 cm. and the height of the cylinderical and conical portions are 10 cm and 6 cm respectively. Find the total surface area of the solid. Solution : The radius of the cone = Height of the cone 6 cm Slant height of the cone 8 cm Radius of cylinder = Radius of a hemisphere Heigt of cylinder = h = 10 cm 10 cm Total Surface area of the Solid = Curved surface areas of Cone + cylinder + hemisphere 8 cm

Exercise - 10.2 3. A Medicine capsule is in the shape of a cylinder with two hemishperes stuck to each of its ends. The length of the capsule is 14 mm. and the width is 5 mm. Find its surface area ? Solution : Width of the capsule = diameter of cylidner = diameter of the two hemispheres = d = 5 mm Radius of capsule = Radius of cylinder = Radius of two hemisphers = r = d/2 = 5/2 = 2.5 mm 5 mm Length of capsule = 14 mm Length of Cylinder = h = 14 – 2(2.5) = 14 – 5 = 9 mm Surface area of the capsule = 2(Curved surface area of hemisphere ) + curved surface area of the cylinder 14 mm 5 mm

Exercise - 10.2 4. Two cubes each of volume 64 cm are joined end to end together . Find the Total surface area of the resulting cuboid ? Solution : Volume of two cubes = 4 òÜ….Ò$. Side of cube 4 òÜ….Ò$. When two cubes are joined end to end together , length of a cuboid = breadth height The Total Surface area of the resulting cuboid

Exercise – 10.2 5. A Storage tank consists of a circular cylinder with a hemisphere stuck on either end . If the external diameter of the cylinder be 1.4 m. and its length be 8 m Find the cost of painting it on the outside at rate of Rs. 20 per m2 . Solution : diameter of cylinder / hemisphere = d = 1.4 m Radius of cylinder / hemisphere = r = d/2 = 1.4/2 = 0.7 m Height of cylinder = h = 8 m Area of the storage tank = 2(Cursed surface area of the hemisphere ) + Curved surface area of the cylinder 8 Ò$ 1.4 Ò$ the cost of painting it on the outside at rate of Rs. 20 per m2 =

Exercise 10.2 6. A sphere , a cylinder and a cone have the same radius. Find the ratio of their curved surface area. Solution: Let radius of sphere,cylinder,cone be r Height of the sphere = r + r = 2 r = Let us assume height of cylinder = height of cone = h = 2r Slant height of the cone Curved surface area of the sphere = Curved surface area of the Cylinder Curved surface area of the cone = Ratio of curved surface area of sphere , cylinder , cone =

Exercise - 10.2 7. A hemisphere is cut out from one face of a cubical wooden block such that the diameter of the hemisphere is equal to the length . Determine the surface area of the remaining Solid. Solution : Let side of the cubical wooden block be a . A hemisphere is cut out from one face of a cubical wooden block Given that the diameter of the hemisphere is equal to the length . There fore d = a Radius of the hemisphere Lateral Surface area of the hemispher The surface area of the remaining Solid = Area of each side of cube + Lateral Surface area of the hemisphere

Exercise 10.2 8. Wooden article was made by scooping out hemisphere from each end of a solid cylinder as shown in the figure. If the height of he cylinder is 10 cm and its radius of base is 3.5 cm . Find the total surface area of the article . Solution : Height of the cylinder = 10 cm, Radius of the base of wooden solid cylinder = r = 3.5 cm Total Surface of the wooden article = Curved surface of the cylinder Curved surface area of the hemisphere

10.2 Surface Area of the combination of Solids