An aquarium with rectangular sides and bottom (and no top) is to hold$\mathbf{5}\mathbf{}\mathit{g}\mathit{a}\mathit{l}$. Find its proportions so that it will use the least amount of material.

Volume of the rectangular aquarium will be $5gal$.

The process of finding the partial derivative of a function is called partial differentiation. In this process, the partial derivative of a function with respect to one variable is found by keeping the other variable constant.

Partial derivative formula:

$$\begin{gathered} {\mathit{f}}_{\mathbf{x}}\mathbf{=}\frac{\mathbf{\partial }\mathbf{f}}{\mathbf{\partial }\mathbf{x}} \\ {\mathit{f}}_{\mathbf{x}}\mathbf{=}\underset{\mathbf{h}\mathbf{\to }\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\frac{\mathbf{f}\left(x+h,y\right)\mathbf{-}\mathbf{f}\left(x,y\right)}{\mathbf{h}} \end{gathered}$$

Let $l\hspace{0.33em},\hspace{0.33em}w\hspace{0.33em},\hspace{0.33em}h$be the dimensions of the aquarium.

$lwh=5$

$h=\frac{5}{I\omega }$ ……. (1)

Now, surface area of the aquarium is given by:

$s=lw+2h(l+w)$

From equation (1) obtain:

$$\begin{gathered} s=lw+2\frac{\left(5\right)}{lw}(l+w)) \\ s=lw+\frac{10}{\omega }+\frac{10}{I} \end{gathered}$$

For minimum value of sas follows:

$$\begin{gathered} \frac{\partial s}{\partial I}=0 \\ \frac{\partial s}{\partial w}=0 \end{gathered}$$

For the width of the aquarium write as follows:

$$\begin{gathered} \frac{ds}{dI}=w-\frac{10}{I^2} \\ \frac{ds}{dI}=0 \\ w-\frac{10}{I^2}=0 \end{gathered}$$

$I^{2}w=10$ …….(2)

Again, calculate as follows:

$$\begin{gathered} \frac{\partial s}{\partial w}=I-\frac{10}{w^2} \\ \frac{\partial s}{\partial w}=0 \\ I-\frac{10}{w^2}=0 \end{gathered}$$

$I{w}^2=10$ …… (3)

From equation (2) and (3):

$$\begin{gathered} \frac{I^{2}w}{I{w}^2}=\frac{10}{10} \\ \frac{I}{w}=1 \end{gathered}$$

$I=w$ ……. (4)

From equation (1) and (4):

$$\begin{gathered} h=\frac{5}{I^2} \\ h=-\frac{5}{\left({\displaystyle \frac{10}{w}}\right)} \\ h=\frac{5w}{10} \end{gathered}$$

$w=2h$ ……. (5)

From equation (4) and (5)$l=w=2h$is obtained.

Therefore, the required proportions of the dimensions are$l=w=2h$.