A milk container of height 16 cm is a frustum with radii 8 cm and 20 cm at the ends. Find the capacity and the cost of milk at Rs 22 per litre that it can hold.
Capacity: \(3328\pi\,\text{cm}^3\approx 10.45\,\text{L}\); Cost ≈ Rs 230
Step 1: Recall the formula for the volume of a frustum of a cone.
\[ V = \dfrac{1}{3} \pi h \big(R^2 + r^2 + Rr\big) \]
Here:
Step 2: Substitute the values into the formula.
\[ V = \dfrac{1}{3} \pi (16) \big(20^2 + 8^2 + 20 \times 8\big) \]
Step 3: Simplify inside the brackets.
\(20^2 = 400\), \(8^2 = 64\), and \(20 \times 8 = 160\).
So, \(400 + 64 + 160 = 624\).
Step 4: Multiply with height.
\(16 \times 624 = 9984\).
Step 5: Multiply with \(\pi/3\).
\[ V = \dfrac{1}{3} \pi (9984) = 3328\pi\,\text{cm}^3 \]
Step 6: Approximate using \(\pi \approx 3.1416\).
\(3328 \times 3.1416 \approx 10452.65\,\text{cm}^3\).
Step 7: Convert to litres (SI unit for capacity).
Since \(1000\,\text{cm}^3 = 1\,\text{L}\),
\(10452.65\,\text{cm}^3 \approx 10.45\,\text{L}\).
Step 8: Find the cost.
Cost per litre = Rs 22.
Total cost = \(10.45 \times 22 \approx 229.9 \approx Rs 230\).
Final Answer: Capacity ≈ 10.45 L, Cost ≈ Rs 230.