A Container Opened at the Top and Made up of a Metal Sheet, is in the Form of a Frustum of a Cone of Height 16 Cm with Radii of Its Lower and Upper Ends as 8 Cm and 20 Cm Respectively. - Mathematics

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Sum

A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container, at the rate of ₹ 50 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹ 10 per 100 cm2. (Take π = 3⋅14)

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Solution

We have to find the cost of milk which can completely fill the container
The volume of container = Volume of a frustum
= `1/3`πh(r12 + r22 + r1r2 )

Here,
height = 16 cm
radius of upper end = 20 cm
And radius of lower end = 8 cm
Plugging the values in the formula we get

Volume of container = `1/3` x 3.14 x 16 ((20)2 + (8)2 + 20 x 8)
= `1/3` x 50.24 (400 + 64 + 160 )
= `1/3` x 50.24 (624)
= 10449.92 cm3        
=10.449 litre                 ....(∵ 1 litre=1000 cm3)

Cost of 1-litre milk is Rs 50
Cost of 10.449 litre milk = 50 x 10.449 = Rs 522.45                                 
We will find the cost of metal sheet to make the container
Firstly, we will find the area of container

Area of container = Curved surface area of the frustum + area of bottom circle    ...(∵ container is closed from bottom)

Area of container = π(r1 + r2)l + πr2
Now, we will find l
l = `sqrt(h^2 + (r_1 − r_2)^2)`

`sqrtl = sqrt((16)^2+(20−8)^2)`

`sqrtl = sqrt((16)^2+(12)^2)`

`sqrtl = sqrt( 256 + 144)`

`sqrtl = sqrt(400)`
`l = 20 cm`
              
Area of frustum = 3.14 x 20(20 + 8)                   
                          = 1758.4 cm2

Area of bottom circle = 3.14 x 82 = 200.96 cm2

Area of container = 1758.4 + 200.96  =1959.36 cm2

Cost of making 100 cm2 = Rs 10
Cost of making 1 cm2 = `10/100` = Rs. 110
Cost of making 1959.36 cm2 = `1/10` x 1959.36 = 195.936                       
Hence, cost of milk is Rs 522.45
And cost of metal sheet is Rs 195.936.

  Is there an error in this question or solution?
2018-2019 (March) 30/4/3

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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:

  1. 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?
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