Question 1. What is the longest pole that can be put in a room of dimensions l = 10 cm, b = 10 cm and h = 5 cm?
Solution: The longest diagonal of a cuboid =
∴ The length of the required pole (diagonal) =
Question 2. Total surface area of a cube is 96 cm2. What is its volume?
Solution: Total surface area of the cube = 6l2
Thus, the volume of the cube = l3 = 43 = 64 cm3
Question 3. The radius of a sphere doubled. What per cent of its volume is increased?
Solution: Original volume = (4/3)πr3
Increased volume = (4/3)π(2r)3 = 32/3πr3
Increase in volume = 32/3πr3 - 4/3πr3 = 28/3πr3
∴ Per cent increase in volume =
Question 4. Write ‘True or False’ for the following statements: (i) A right circular cylinder just encloses a sphere of radius r as shown in the figure. The area of the sphere is equal to the curved surface area of the cylinder.
∵ [Radius of the sphere] = [Radius of the cylinder] = r
∴ Diameter of the sphere = 2r
⇒ Height of the cylinder (h) = 2r
Now, surface area of the sphere = 4πr2
And curved surface area of the cylinder = 2πrh = 2πr (2r) = 4πr2
(ii) An edge of a cube measures ‘r’ cm. If the largest possible right circular cone is cut out of this cube, then the volume of the cone (in cm3) is 1/6πr3
∵ Height of the cone = r cm
∴ Diameter of the base of the cone = r cm
⇒ Radius of the base of the cone = (r/2) cm
Now volume of the cone
Question 5. If the total surface area of a sphere is 154 cm2. Find its total volume.
Solution: Let ‘r’ be radius of the sphere
∴ Total S.A. = 4 π r2 = 154 cm2
Question 6. If the radius of a sphere is 3r then what is its volume?