Find the most economical proportions for a tent as in the figure, with no floor.

The given tent is without floor.

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:

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

In the figure the tent can be divided into two parts

The top part as pup tent and another one is cuboid.

Volume is given as follows:

$$\begin{gathered} V=\frac{1}{2}bhl \\ V=z{x}^2\tan \theta \end{gathered}$$

Area is given as follows:

$A=2x^2\tan \theta +\frac{2\pi x}{\cos \theta }$

Since $x,z,\theta$are independent $A=2x\tan \theta +\frac{2V}{x\cos \theta \tan \theta }$.

Now two independent variables to minimize $\frac{\partial A}{\partial x}=0$and $\frac{\partial A}{\partial \theta }=0$.

$$\begin{gathered} \frac{\partial A}{\partial x}=4x\tan \theta -\frac{2V\cos ec\theta }{x^2} \\ \frac{\partial A}{\partial x}=0 \end{gathered}$$

And calculate as follows:

$$\begin{gathered} \frac{\partial A}{\partial \theta }=2x^2{\sec }^2\theta -\frac{2V\cos ec\theta \cot \theta }{x} \\ \frac{\partial A}{\partial \theta }=0 \end{gathered}$$

Equate the above two equations, obtain:

$\begin{array}{rcl}\frac{V\cos ec\theta }{2\tan \theta }& =& \frac{V\cos ec\theta \cot \theta }{{\sec }^2\theta }\\ {\cos }^2\theta & =& \frac{1}{2}\\ \theta & =& \frac{\pi }{4}\end{array}$

Hence,$V=X^{2}Z$.

$$\begin{gathered} \frac{\partial A}{\partial x}=2x \\ \frac{\partial A}{\partial x}=z\sqrt{2} \end{gathered}$$

The high of the upper part of a tent is$x=\frac{1}{2}$.

Now for rectangular part (lower part) without floor

Area is $A=4xy+4xz+zy$.