CBSE Class 10 Surface Areas and Volumes Sure Shot Questions Set 16

Vijay has rain water harvesting plant on his roof. After rain, all the water that is collected on the roof of \( 22 \text{ m} \times 20 \text{ m} \) is drained into the cylindrical tank having diameter of base \( 2 \text{ m} \) and height \( 3.5 \text{ m} \). It rained heavily last night and in the morning the tank is just full.

Question. How much water is collected in the tank (in litres)?
Answer: We have,
Radius of cylindrical tank, \( r = \frac{2}{2} = 1 \text{ m} \)
Height of cylindrical tank, \( h = 3.5 \text{ m} \)
Volume of cylindrical tank = \( \pi r^2 h \)
\( = \frac{22}{7} \times (1)^2 \times 3.5 \)
\( = 11 \text{ m}^3 \)
Since, \( 1 \text{ m}^3 = 1000 \text{ l} \)
\( 11 \text{ m}^3 = 11000 \text{ litres} \).
\( 11000 \text{ litres} \) of water is collected in tank.

Question. Find the rainfall in cm ?
Answer: Let the rainfall be \( x \text{ m} \),
\( \therefore \) Volume of water = Volume of cuboid
\( = 22 \text{ m} \times 20 \text{ m} \times x \text{ m} \).
\( = (22 \times 20 \times x) \text{ m}^3 \)
\( = 440x \text{ m}^3 \)
Since, the tank is just full of water that drains out of the roof into the tank.
\( \therefore \) Volume of water = Volume of the cylindrical vessel
\( \Rightarrow 22 \times 20 \times x = 11 \)
\( x = \frac{11}{440} = \frac{1}{40} \text{ m} \)
\( = \frac{100}{40} \text{ cm} = 2.5 \text{ cm} \)

A wooden toy is in the shape of a right circular cylinder with hemisphere on one end and a cone on the other. The height and radius of cylinder part are \( 13 \text{ cm} \) and \( 5 \text{ cm} \) respectively. The radii of the hemispherical and conical parts are same as that of the cylindrical part. Height of the conical part is \( 12 \text{ cm} \). The conical portion and the hemispherical portion is to be painted green and cylindrical portion is to be painted red. Based on the given information, answer the following questions.

Question. How much area of the toy is painted with red colour ?
Answer: Required area = Curved surface area of cylinder
\( = 2\pi rh \)
Here, \( h = 13 \text{ cm} \), \( r = 5 \text{ cm} \).
\( = 2 \times \frac{22}{7} \times 5 \times 13 \)
\( = \frac{2860}{7} \text{ cm}^2 \)

Question. How much area of the toy is painted with green colour ?
Answer: Required area = Curved surface area of cone + Curved surface area of hemisphere.
\( = \pi rl + 2\pi r^2 \)
Here, \( r = 5 \text{ cm} \), \( h = 12 \text{ cm} \)
and \( l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = 13 \text{ cm} \)
\( = \frac{22}{7} \times 5 \times 13 + 2 \times \frac{22}{7} \times 5 \times 5 \)
\( = \frac{1430}{7} + \frac{1100}{7} = \frac{2530}{7} \text{ cm}^2 \)

Question. Find the total surface area of the toy.
Answer: Total surface area of toy \( = \frac{2860}{7} + \frac{2530}{7} \)
\( = \frac{5390}{7} = 770 \text{ cm}^2 \)

A thirsty crow saw a spherical glass vessel with cylindrical neck \( 8 \text{ cm} \) long and \( 2 \text{ cm} \) in diameter. The diameter of spherical part is \( 8.5 \text{ cm} \). The vessel is half filled with water. But due to long neck of vessel crow was not able to drink water from it. He saw few spherical marbles of diameter \( 1 \text{ cm} \) lying near by the vessel. He start dropping them inside the vessel one by one. Based on the given information, answer the following questions :

Question. What is the total volume of the vessel ?
Answer: We have,
Radius of cylindrical neck (\( r \)) \( = \frac{2}{2} \text{ cm} = 1 \text{ cm} \)
Length of cylindrical neck (\( h \)) \( = 8 \text{ cm} \)
Radius of spherical part (\( R \)) \( = \frac{8.5}{2} = 4.25 \text{ cm} \)
Volume of cylindrical part \( = \pi r^2 h = \pi \times (1)^2 \times 8 = 8\pi \text{ cm}^3 \)
Volume of spherical part \( = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (4.25)^3 = 102.354\pi \text{ cm}^3 \)
Total volume of vessel = Volume of cylindrical part + Volume of spherical part
\( = 8\pi + 102.354\pi \text{ cm}^3 = 110.354\pi \text{ cm}^3 \)

Question. How many marbles crow has to drop inside the vessel to drink the water ?
Answer: Since, vessel is half filled.
\( \therefore \) Volume of water to be filled \( = \frac{110.354\pi}{2} \text{ cm}^3 = 55.177\pi \text{ cm}^3 \)
Volume of each marble of radius \( \left( \frac{1}{2} \text{ cm} \right) = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left( \frac{1}{2} \right)^3 \text{ cm}^3 = \frac{\pi}{6} \text{ cm}^3 \)
No. of marbles \( = \frac{\text{Volume of water to be filled}}{\text{Volume of each marble}} = \frac{55.177\pi}{\frac{\pi}{6}} = 331.06 \approx 331 \)
Crow has to drop \( 331 \) marbles so it can drink water from it.

Question. If the diameter of a sphere is \( 14 \text{ cm} \), then what is its curved surface area ?
Answer: \( 616 \text{ cm}^2 \)

Question. The radius and height of a right-circular cone are \( 12 \text{ cm} \) and \( 9 \text{ cm} \) respectively. What is the curved surface area of that cone ?
Answer: \( 180\pi \text{ cm}^2 \)

Question. If the diameter of a sphere is \( 14 \text{ cm} \), then what is its curved surface area ?
Answer: \( 616 \text{ cm}^2 \)

Question. The radius and height of a right-circular cone are \( 12 \text{ cm} \) and \( 9 \text{ cm} \) respectively. What is the curved surface area of that cone ?
Answer: \( 565.71 \text{ cm}^2 \)

Question. The volume of a right-circular cylinder is \( 352 \text{ cm}^3 \) and the height is \( 7 \text{ cm} \). Find the radius of the base.
Answer: \( 4 \text{ cm} \)

Question. If the radii of a sphere and a right-circular cylinder are \( 3 \text{ cm} \) each and if their volumes are equal as well, then find the height of the cylinder.
Answer: \( 4 \text{ cm} \)

Question. The curved surface area and the volume of a right circular cylinder are numerically equal. Find the radius of the cylinder.
Answer: \( 2 \text{ cm} \)

Question. The volumes of a sphere and cylinder are equal. The diameter of the base of the cylinder is equal to the diameter of the sphere. Find the ratio between the radius of the base and the height of the cylinder.
Answer: \( 3 : 4 \)

Question. \( 77 \text{ m}^2 \) of canvas is required to make a conical tent of slant height \( 7 \text{ m} \). Find the area of the base.
Answer: \( \frac{77}{2} \text{ m}^2 \)

Question. The volume of the right-circular cone of height \( 24 \text{ cm} \) is \( 1232 \text{ cm}^3 \). Find the lateral surface area of the cone.
Answer: \( 550 \text{ cm}^2 \)

Question. Two cubes each of volume \( 64 \text{ cm}^3 \) are joined end to end to form a solid. Find the surface area and volume of the resulting cuboid. 
Answer: \( 160 \text{ cm}^2, 128 \text{ cm}^3 \)

Question. Metallic spheres of radii \( 6 \text{ cm}, 8 \text{ cm} \) and \( 10 \text{ cm} \) respectively are melted to form a single solid sphere. Find the radius of the resulting sphere. 
Answer: \( 12 \text{ cm} \)

Question. A \( 20 \text{ m} \) deep well with diameter \( 7 \text{ m} \) is dug and the earth from digging is evenly spread out to form a platform of \( 22 \text{ m} \) by \( 14 \text{ m} \). Find the height of the platform. 
Answer: \( 2.5 \text{ m} \)

Question. A copper wire \( 3 \text{ mm} \) in diameter is wound about a cylinder whose length is \( 1.2 \text{ m} \) and diameter \( 10 \text{ cm} \), so as to cover the curved surface of the cylinder. Find the length and mass of the wire, assuming the density of the copper wire to be \( 8.88 \text{ g/cm}^3 \). 
Answer: \( 125.6 \text{ m}, 111 \text{ kg } 532 \text{ g} \)

Question. Selvi’s house has an overhead tank in the shape of a cylinder. It is filled up by pumping water from an underground tank that is cuboid in shape. The dimensions of the cuboid are \( 1.57 \text{ m} \times 1.44 \text{ m} \times 0.95 \text{ m} \). The radius of the overhead tank is \( 60 \text{ cm} \) and its height is \( 95 \text{ cm} \). Find the height of the water-level in the underground tank after the overhead tank has been filled up completely. Compare the capacities of both the tanks. (Use \( \pi = 3.14 \)) 
Answer: \( 47.5 \text{ cm}, 2 : 1 \)

Question. How many coins \( 1.75 \text{ cm} \) in diameter and \( 2 \text{ mm} \) thickness must be melted to form a cuboid of dimensions \( 5.5 \text{ cm} \times 10 \text{ cm} \times 3.5 \text{ cm} \)? 
Answer: \( 400 \)

Question. Water in a canal, \( 6 \text{ m} \) deep and \( 1.5 \text{ m} \) wide is flowing at a speed of \( 10 \text{ km/hr} \). How much area will it irrigate in \( 30 \text{ minutes} \), if \( 8 \text{ cm} \) of standing water is needed for irrigation ? 
Answer: \( 562500 \text{ m}^2 \)

Question. A farmer connects a pipe of internal diameter \( 20 \text{ cm} \) from a canal into a cylindrical tank which is \( 10 \text{ m} \) in diameter and \( 2 \text{ m} \) deep. If the water flows through the pipe at the rate of \( 3 \text{ km/hr} \) then in how much time will the tank be completely filled ? 
Answer: \( 1 \text{ hour and } 40 \text{ minutes} \)

Question. A wooden toy rocket is in the shape of a cone mounted on a cylinder. The height of the entire rocket is \( 26 \text{ cm} \) while the height of the conical part is \( 6 \text{ cm} \). The base of the conical portion has a diameter of \( 5 \text{ cm} \) while the diameter of the cylinder is \( 3 \text{ cm} \). If the conical portion is to be painted orange and the cylindrical portion yellow, find the area of the rocket painted with each of these colours. [Use \( \pi = 3.14 \)]
Answer: \( 195.47 \text{ cm}^2 \)

Question. A wooden toy was made from the rest of the solid cylinder after scooping out a hemisphere of same radius from each of it. If the height of the cylinder is \( 10 \text{ cm} \) and its base radius is \( 3.5 \text{ cm} \), find the total surface area. [Use \( \pi = \frac{22}{7} \)]
Answer: \( 374 \text{ cm}^2 \)

Question. A tent is in the form of a right-circular cylinder of base diameter \( 4 \text{ m} \) and height \( 2.1 \text{ m} \) surmounted by a right circular cone of the same base radius and slant height \( 2.8 \text{ cm} \). Find the area of the canvas used and the cost of canvas at ₹ \( 500 \) per square metre. 
Answer: ₹ \( 22000 \)

Question. A solid is in the shape of a cone mounted on a hemisphere, the radius of each of them being \( 1 \text{ cm} \) and the total height of the cone equal to its radius. Find the volume of the solid in terms of \( \pi \). 
Answer: \( \pi \text{ cm}^3 \)