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A milk container is made of metal sheet in the shape of frustum of a cone whose volume is 1045937cm3. The radii of its lower and upper circular ends are 8cm and 20cm respectively. Find the cost of the metal sheet used in making the container at the rate of Rs.1.40 per cm3.(Use π=22.7).

Answer
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Hint: At first use the volume formula π3h(R2+r2+Rr) which is volume of frustum to get the value of ‘h’ or height. Now further use the data to get value of surface area using π(R+r)(Rr)2+h2+πR2 and then multiply with Rs.1.40 to find the cost.

Complete step-by-step answer:
In the question we are given a milk container which is in the shape of a frustum of a cone. We have to find the total surface area of the cone so as to get the cost of the metal sheet to cover the container.

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The data or the dimensions of the frustum of the cone given are radii of its lower and upper circular surfaces are 8cm and 20cm respectively. The volume is already given as 1045937cm3.

To find the surface area we are given the formula with opening on the smaller side of base ‘r’ of frustum,

π(R+r)(Rr)2+h2+πR2

Where R, r are the radius of larger and smaller circular bases respectively and h is height.
The formula of volume is,

π3h(R2+r2+Rr)

Where R, r are the radius of larger and smaller circular bases respectively and h is height.

The volume given is 1045937cm3 which can be written as 732167cm3.

Now it can be written as,

π3h(R2+r2+Rr)=732167cm3

Here π=227,R=20,r=8.

So,

13×227×h(202+82+20×8)=732167

Which can be written as,

h(202+82+160)=732167×2122

Which can be further solved as,

h(624)=9984

So,

h=9984624=16cm

Hence the height of the frustum of the cone is 16cm.
Now, we have to get the value of surface area of frustum with an opening on the circular base of radius ‘r’,

π(R+r)(Rr)2+h2+πR2

Here R=20,r=8,h=16 & π=227

So, the surface area is,

=227×{(20+8)(208)2+162+202}

Which can be calculated as,

227×28×122+162+227×(20)2

By doing calculations carefully we can write it as,

227×{28×400+202}=227×{560+4}=211207cm2

Hence, the area of metal sheet required is 211207cm2

Now for the costing it was given that for 1cm2 of metal material Rs.1.40 is required.

So, the total cost is Rs.(211207×1.40) which is equal to Rs. 4224.

Hence the cost of metal sheet is Rs. 4224.


Note: Students might get confused in this problem whether to include πr2 in the surface area and costing or not. They should not include it because in the question we are given to find the area of the container which is opened in the circular base with a smaller radius.