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Question
A solid metallic sphere of diameter 28 cm is melted and recast into a number of smaller cones, each of diameter 4 \[\frac{2}{3}\] cm and height 3 cm. Find the number of cones so formed.
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Solution
The radius of solid metallic sphere, `R= 28/2`= 14 cm
The volume of sphere
`= 4/3 pi R^3`
`= 4/3 xx pi xx (14)^3`
`= 4/3pi xx 14 xx 14 xx 14`
`= (10976 pi)/3 cm^3`
Given, the sphere is recast into smaller cones.
The radius of cone,
`r = 14/(3 xx 2)`
`= 7/3 cm`
The height of cone h = 3 cm
Let n be the no. of smaller cones.
Clearly, the volume of solid sphere = n × volume of one smaller cone
\[\frac{10976}{3}\pi = n \times \frac{1}{3}\pi \times \left( \frac{7}{3} \right)^2 \times 3\]
\[n \times \frac{49}{3} = 10976\]
\[n = \frac{10976 \times 3}{49}\]
\[n = 672\]
Thus, the no. of smaller cones = 672
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Khurja is a city in the Indian state of Uttar Pradesh famous for the pottery. Khurja pottery is traditional Indian pottery work which has attracted Indians as well as foreigners with a variety of tea sets, crockery and ceramic tile works. A huge portion of the ceramics used in the country is supplied by Khurja and is also referred as "The Ceramic Town". One of the private schools of Bulandshahr organised an Educational Tour for class 10 students to Khurja. Students were very excited about the trip. Following are the few pottery objects of Khurja.
Students found the shapes of the objects very interesting and they could easily relate them with mathematical shapes viz sphere, hemisphere, cylinder etc. |
Maths teacher who was accompanying the students asked the following questions:
- The internal radius of hemispherical bowl (filled completely with water) in I is 9 cm and the radius and height of the cylindrical jar in II are 1.5 cm and 4 cm respectively. If the hemispherical bowl is to be emptied in cylindrical jars, then how many cylindrical jars are required?
- If in the cylindrical jar full of water, a conical funnel of the same height and same diameter is immersed, then how much water will flow out of the jar?
A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of the cylindrical part are 3 m and 14 m respectively, and the total height of the tent is 13.5 m, find the area of the canvas required for making the tent, keeping a provision of 26 m2 of canvas for stitching and wastage. Also, find the cost of the canvas to be purchased at the rate of ₹ 500 per m2.
