A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The total length is 14 mm, and the diameter is 5 mm. Find its volume and surface area. If the cost of covering the capsule with edible coating is ₹5 per 100 mm², what will be the total cost of manufacturing 500 such capsules?

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Points to Remember:

  • Volume and surface area calculations for a composite 3D shape (cylinder + hemispheres).
  • Cost calculation based on surface area and unit cost.
  • Application of geometric formulas.

Introduction:

This question involves calculating the volume and surface area of a capsule-shaped object, which is a combination of a cylinder and two hemispheres. We will use standard geometric formulas to determine these values and then calculate the total cost of manufacturing 500 such capsules, given a specified coating cost per unit area. This problem combines geometry with practical application in manufacturing cost estimation.

Body:

1. Calculating the Volume:

  • Cylinder Volume: The length of the cylindrical part is 14 mm – 2(radius) = 14 mm – 2(2.5 mm) = 9 mm (since diameter = 5 mm, radius = 2.5 mm). The volume of a cylinder is πr²h, where r is the radius and h is the height. Therefore, the cylinder’s volume is π*(2.5 mm)² * 9 mm ≈ 176.71 mm³.

  • Hemisphere Volume: Each hemisphere has a radius of 2.5 mm. The volume of a sphere is (4/3)πr³, so the volume of one hemisphere is (1/2)(4/3)π(2.5 mm)³ ≈ 32.72 mm³. The total volume of two hemispheres is approximately 65.44 mm³.

  • Total Volume: The total volume of the capsule is the sum of the cylinder’s volume and the two hemispheres’ volumes: 176.71 mm³ + 65.44 mm³ ≈ 242.15 mm³.

2. Calculating the Surface Area:

  • Cylinder Surface Area: The curved surface area of a cylinder is 2πrh = 2π(2.5 mm)(9 mm) ≈ 141.37 mm².

  • Hemisphere Surface Area: The surface area of a sphere is 4πr², so the surface area of one hemisphere is (1/2)*4π(2.5 mm)² ≈ 19.63 mm². The total surface area of two hemispheres is approximately 39.26 mm².

  • Total Surface Area: The total surface area of the capsule is the sum of the cylinder’s curved surface area and the two hemispheres’ surface areas: 141.37 mm² + 39.26 mm² ≈ 180.63 mm².

3. Calculating the Total Manufacturing Cost:

  • Cost per capsule: The cost of coating one capsule is (180.63 mm² / 100 mm²) * ₹5 = ₹9.03 (approximately).

  • Total cost for 500 capsules: The total cost of manufacturing 500 capsules is 500 * ₹9.03 = ₹4515.

Conclusion:

The volume of a single medicine capsule is approximately 242.15 mm³, and its surface area is approximately 180.63 mm². Based on a coating cost of ₹5 per 100 mm², the total cost of manufacturing 500 capsules is approximately ₹4515. This calculation demonstrates a practical application of geometric principles in industrial cost estimation. Further optimization could involve exploring alternative capsule designs or negotiating better coating material prices to reduce manufacturing costs while maintaining product quality and efficacy. This approach emphasizes efficiency and cost-effectiveness in pharmaceutical production, aligning with principles of sustainable business practices.

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