Two cubes are joined together. Which of the following will be true about the resulting solid? i)Length of the solid will remain same. ii) Breadth of the solid will become twice. iii) Height of the solid will become twice. iv) Length of the solid will become twice.
Answers
Step-by-step explanation:
Exercise 13.1
Q.1. Two cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.
Sol. Volume of each cube = 64 cm3
∴ Total volume of the two cubes = 2 × 64 cm3
= 128 cm3
Let the edge of each cube = x
∴ x3 = 64 = 43
∴ x = 4 cm
Now, Length of the resulting cuboid l = 2x cm
Breadth of the resulting cuboid b = x cm
Height of the resulting cuboid h = x cm
∴ Surface area of the cuboid = 2 (lb + bh + hl) = 2[(2x . x) + (x . x) + (x . 2x)]
= 2[(2 × 4 × 4) + (4 × 4) + (4 × 2 × 4)] cm2
= 2 [32 + 16 + 32] cm2 = 2[80] cm2 = 160 cm2.
Q.2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Sol. For cylindrical part:
Radius (r) = 7 cm
Height (h) = 6 cm
∴ Curved surface area
= 2πrh
For hemispherical part:
Radius (r) = 7 cm
∴ Surface area = 2πr2
∴ Total surface area
= (264 + 308) cm2 = 572 cm2.
Q.3. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
Sol. Here, r = 3.5 cm
∴ h = (15.5 – 3.5) cm = 12.0 cm
Surface area of the conical part
= πrl
Surface area of the hemispherical part
= 2πr2
∴ Total surface area of the toy
= πrl + 2πr2 = πr (l + 2r) cm2
∵ l2 = (12)2 + (3.5)2
Q.4. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
Sol. Side of the block = 7 cm
⇒ The greatest diameter of the hemisphere = 7 cm
Surface area of the solid
= [Total S.A. of the cubical block] + [S.A. of the hemisphere] – [Base area of the hemisphere]
= (6 × l2) + 2πr2 – πr2
Q.5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
Sol. Let ‘l’ be the side of the cube.
∴ The greatest diameter of the curved hemisphere = l
Q.6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig.). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
Sol. Radius of the hemispherical part
∴ Surface area of one hemispherical part = 2πr2
Q.7. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of Rs. 500 per m2. (Note that the base of the will not be covered with canvas.)
Sol. For cylindrical part:
Q.8. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm2.
Sol. For cylindrical part:
Height = 2.4 cm
Diameter = 1.4 cm
⇒ Radius (r) = 0.7 cm
⇒ Total surface area of the cylindrical part
Q.9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as show in Fig. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.
Sol. Radius of the cylinder (r) = 3.5 cm
Height of the cylinder (h) = 10 cm
∴ Total surface area = 2πrh + 2πr2 = 2πr(h +r)
Answer:
When two cubes are joined together, length of the solid will become twice. Frankly That is true as the length of the solid will be twice the side length of one of the cubes.
Hence the correct option is iv).
Step-by-step explanation:
The resulting solid formed by joining two cubes will be a cuboid. Let's assume that each cube has side length "a".
The resulting cuboid will have length, breadth, and height as follows:
Length = 2a (the length of the resulting cuboid will be equal to the combined length of two cubes)
Breadth = a (the breadth of the resulting cuboid will be equal to the side length of one of the cubes)
Height = a (the height of the resulting cuboid will be equal to the side length of one of the cubes)
So, among the given options:
i) Length of the solid will remain same. - This is false as the length of the solid will be twice the side length of one of the cubes.
ii) Breadth of the solid will become twice. - This is false as the breadth of the solid will remain the same as the side length of one of the cubes.
iii) Height of the solid will become twice. - This is false as the height of the solid will remain the same as the side length of one of the cubes.
iv) Length of the solid will become twice. - This is true as the length of the solid will be twice the side length of one of the cubes.
Therefore, the correct option is iv) Length of the solid will become twice.
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