A solid metallic hemisphere of radius 8 cm is melted and recast into a right circular cone of base radius 6 cm. Determine the height of the cone.
\(\displaystyle h=\dfrac{256}{9}\,\text{cm}\approx 28.44\,\text{cm}\)
A rectangular water tank of base \(11\,\text{m}\times 6\,\text{m}\) contains water up to a height of 5 m. If the water is transferred to a cylindrical tank of radius 3.5 m, find the height of water in the cylinder.
\(\displaystyle h=\dfrac{60}{7}\,\text{m}\approx 8.57\,\text{m}\)
How many cubic centimetres of iron are required to construct an open box whose external dimensions are 36 cm, 25 cm and 16.5 cm, the thickness being 1.5 cm? If 1 cm³ of iron weighs 7.5 g, find the weight of the box.
Iron volume = 3960 cm³; Weight = 29.7 kg
A fountain-pen barrel is a cylinder of length 7 cm and diameter 5 mm. A full barrel writes 3300 words on average. How many words can be written with a bottle containing \(\dfrac15\) litre of ink?
4,80,000 words
Water flows at \(10\,\text{m min}^{-1}\) through a cylindrical pipe of diameter 5 mm. How long to fill a conical vessel of diameter 40 cm and depth 24 cm?
\(51.2\) minutes
A heap of rice is a cone of diameter 9 m and height 3.5 m. Find the volume of rice and the canvas required to just cover it.
Volume: \(\displaystyle \dfrac{23.625\pi}{1}\,\text{m}^3\approx 74.2\,\text{m}^3\); Canvas area: \(\pi r l=\pi\cdot4.5\cdot\sqrt{4.5^2+3.5^2}\approx 80.6\,\text{m}^2\).
A factory makes 1,20,000 pencils daily. Each pencil is a cylinder of length 25 cm and base circumference 1.5 cm. Find the cost of colouring the curved surfaces at Rs 0.05 per dm².
Rs 2,250
Water flows at 15 km/h through a pipe of diameter 14 cm into a cuboidal pond \(50\,\text{m}\times44\,\text{m}\). In what time will the water level rise by 21 cm?
2 hours
A solid iron cuboid \(4.4\,\text{m}\times2.6\,\text{m}\times1\,\text{m}\) is recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.
112 m
500 persons take a dip in a cuboidal pond \(80\,\text{m}\times50\,\text{m}\). If the average water displacement per person is \(0.04\,\text{m}^3\), find the rise in water level.
0.5 cm
Sixteen glass spheres, each of radius 2 cm, are packed into a cuboidal box of internal dimensions \(16\,\text{cm}\times8\,\text{cm}\times8\,\text{cm}\) and the box is then filled with water. Find the volume of water filled.
\(\displaystyle 1024-\dfrac{512}{3}\pi\;\text{cm}^3\;\approx 488\,\text{cm}^3\)
A milk container of height 16 cm is a frustum with radii 8 cm and 20 cm at the ends. Find the capacity and the cost of milk at Rs 22 per litre that it can hold.
Capacity: \(3328\pi\,\text{cm}^3\approx 10.45\,\text{L}\); Cost ≈ Rs 230
A cylindrical bucket (height 32 cm, base radius 18 cm) is filled with sand and emptied to form a conical heap of height 24 cm. Find the radius and slant height of the heap.
Radius = 36 cm, Slant height = \(\sqrt{36^2+24^2}=\sqrt{1872}\approx 43.3\,\text{cm}\)
A rocket is a cylinder (radius 3 cm, height 12 cm) surmounted by a cone of the same radius and slant height 5 cm. Find the total surface area and volume. [Use \(\pi=3.14\)].
TSA = \(96\pi\approx 301.44\,\text{cm}^2\); Volume = \(120\pi\approx 376.8\,\text{cm}^3\)
A building is a cylinder surmounted by a hemispherical dome and contains \(41\dfrac{19}{21}\,\text{m}^3\) of air. If the internal diameter of the dome equals the total height above the floor, find the height of the building.
4 m
A hemispherical bowl of internal radius 9 cm is full of liquid. It is to be filled into cylindrical bottles each of radius 1.5 cm and height 4 cm. How many bottles are needed?
54 bottles
A solid cone (height 120 cm, radius 60 cm) is placed in a right circular cylinder full of water of height 180 cm such that it touches the bottom. The cylinder radius equals that of the cone. Find the volume of water left in the cylinder.
\(\displaystyle 5.04\times10^5\pi\,\text{cm}^3\)
Water flows through a pipe (inner radius 1 cm) at 80 cm/s into an empty cylindrical tank of radius 40 cm. What is the rise in water level in half an hour?
90 cm
Rain from a roof \(22\,\text{m}\times20\,\text{m}\) drains into a cylindrical vessel of diameter 2 m and height 3.5 m. If the vessel is just filled, find the rainfall in cm.
2.5 cm
A wooden pen stand is a cuboid \(10\times5\times4\,\text{cm}\) with four conical depressions (radius 0.5 cm, depth 2.1 cm) and a cubical depression of edge 3 cm. Find the volume of wood in the stand.
\(\displaystyle 173-0.7\pi\;\text{cm}^3\approx 170.8\,\text{cm}^3\)