The diagram shows a solid object. It is made up of a cylinder with a hemisphere above it. The cylind

Question

The diagram shows a solid object. It is made up of a cylinder with a hemisphere above it.
The cylinder has a radius of $x$ cm and a height of $y$ cm. The radius of the hemisphere is $2x$ cm.

Express $y$ in terms of $x$ if the total volume of the solid is $72\pi$ $\text{cm}^3$.[2]
Show that the total surface area, $A$ $\text{cm}^2$, of the solid is $144\pi\left(\frac{1}{x} + \frac{x^2}{108}\right)$.[2]
Find the stationary value of the algebraic expression for $A$ and its nature. Check whether it is attainable by the solid with positive cylinder height. Describe the least possible surface-area bound.[4]

Accepted Answer

$y=72/x^2-16x/3$. The algebraic stationary point $x=\sqrt[3]{54}$ is outside the physical domain. For positive cylinder height there is no attained minimum; infimum $12\pi(27/2)^{2/3}\text{ cm}^2$.

2024 GESS PRELIMS P2 Q9

2024 GESS PRELIMS P2 Q9

Solution:

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