The diagram shows a cuboid water tank with no lid, where the base rectangle has dimensions of $3x$

Question

The diagram shows a cuboid water tank with no lid, where the base rectangle has dimensions of $3x$ cm by $x$ cm. The tank has a height of $h$ cm and has a fixed volume of $3000$ cm$^3$. Assume that the thickness of the tank is negligible.

Show that the total surface area of the tank, $A$ cm$^2$, of the tank, is given by
$A = 3x^2 + \frac{8000}{x}.$
Given that $x$ can vary, find the values of $x$ and $h$ for which $A$ has a stationary value.
Hence determine the nature of this stationary value and find its value.

Accepted Answer

(i) $A=3x^2+\dfrac{8000}{x}$ (ii) $x=10\sqrt[3]{\dfrac43}\text{ cm},\ h=\dfrac{15}{2}\sqrt[3]{\dfrac43}\text{ cm}$ (iii) Minimum area $=900\sqrt[3]{\dfrac{16}{9}}\text{ cm}^2$

Differentiation - Problems Involving Maximisation and Minimisation

Differentiation - Problems Involving Maximisation and Minimisation

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