A milk container is made of metal sheet in the shape of frustum of a cone whose volume is $10459 \frac{3}{7}$ cm³. The radii of its lower and upper circular ends are 8 cm and 20 cm, respectively. Find the cost of metal sheet used in making the container at the rate of ₹1.40 per cm².
[CBSE 2010]

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Solution

$We have, Radius of the upper end, R = 20 cm and Radius of the lower end, r = 8 cm Let the height of the container be h . As, Volume of the container = 10459 \frac{3}{7} {cm}^{3} \Rightarrow \frac{1}{3} π h (R^{2} + r^{2} + R r) = \frac{73216}{7} \Rightarrow \frac{1}{3} \times \frac{22}{7} \times h \times (20^{2} + 8^{2} + 20 \times 8) = \frac{73216}{7} \Rightarrow \frac{22 h}{21} \times (400 + 64 + 160) = \frac{73216}{7} \Rightarrow \frac{22 h}{21} \times 624 = \frac{73216}{7} \Rightarrow h = \frac{73216 \times 21}{7 \times 22 \times 624} \Rightarrow h = 16 cm Also, The slant height of the container, l = \sqrt{{(R - r)}^{2} + h^{2}} = \sqrt{{(20 - 8)}^{2} + 16^{2}} = \sqrt{12^{2} + 16^{2}} = \sqrt{144 + 256} = \sqrt{400} = 20 cm Now, Total surface area of the container = π l (R + r) + π r^{2} = \frac{22}{7} \times 20 \times (20 + 8) + \frac{22}{7} \times 8 \times 8 = \frac{22}{7} \times 20 \times 28 + \frac{22}{7} \times 64 = \frac{22}{7} \times (560 + 64) = \frac{22}{7} \times 624 = \frac{13728}{7} {cm}^{2} So, the cost of metal sheet = 1.4 \times \frac{13728}{7} = ₹ 2745.60$

Hence, the cost of the metal sheet used for making the milk container is ₹2745.60.

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A milk container is made of metal sheet in the shape of frustum of a cone whose volume is 1045937 cm3. The radii of its lower and upper circular ends are 8 cm and 20 cm, respectively. Find the cost of metal sheet used in making the container at the rate of ₹1.40 per cm2. [CBSE 2010]

A milk container is made of metal sheet in the shape of frustum of a cone whose volume is $10459 \frac{3}{7}$ cm³. The radii of its lower and upper circular ends are 8 cm and 20 cm, respectively. Find the cost of metal sheet used in making the container at the rate of ₹1.40 per cm².
[CBSE 2010]