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A manufacturing company produces airtight cylindrical containers for storing sensitive chemicals…
A manufacturing company produces airtight cylindrical containers for storing sensitive chemicals. For safety, each container is placed inside a spherical protective shell of radius $r$ cm. The cylindrical container is so designed that it fits completely inside the sphere, with its axis passing through the centre of the sphere. The company wants to utilise the space better and wants the cylindrical container to occupy the maximum possible volume inside the spherical shell.
To achieve this, the design team applies the concept of Maxima and Minima to determine the optimal dimensions of the cylindrical container.
Answer the following questions based on the above situation:
Answer the following questions based on the above situation:
(i)[1.0]
Obtain a relation between the height $h$ and the radius $x$ of the cylinder.
(ii)[1.0]
Write the expression for the volume $V$ of the cylindrical container in terms of $x$ and $r$.
(iii)[3.0]
Find the value of $x$ for which the volume of the cylinder is maximum.
(iv)[1.0]
Hence, find the maximum volume of the cylindrical container.
Answer
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From ISC 2026 Improvement Mathematics Paper 1, question 11.