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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 cm^{2}. (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(r_{1}^{2 }+ r_{2}^{2 }+ r_{1}r_{2 })

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 cm^{3}

=10.449 litre ....(∵ 1 litre=1000 cm^{3})

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(r_{1 }+ r_{2})l + πr^{2}

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 cm^{2}

Area of bottom circle = 3.14 x 8^{2 }= 200.96 cm^{2}

Area of container = 1758.4 + 200.96 =1959.36 cm^{2}

Cost of making 100 cm^{2} = Rs 10

Cost of making 1 cm^{2 }= `10/100` = Rs. 110

Cost of making 1959.36 cm^{2 }= `1/10` x 1959.36 = 195.936

Hence, cost of milk is Rs 522.45

And cost of metal sheet is Rs 195.936.

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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?