Question

For a closed rectangular box, with a square base x by x cm and height h cm, find the dimensions giving the minimum surface area, given that the volume is 7 cm3. Enclose arguments of functions, numerators, and denominators in parentheses. For example, sin(2x) or (a−b)/(1+n).

Answer 1

Answer:

The dimensions are  $x=\sqrt[3]{7}$ and $h=\sqrt[3]{7}$.

Step-by-step explanation:

Consider the provided information.

The volume is 7 cm³.

The volume of box is: $V=lbh$

Since the base is square (x by x cm) therefore the volume is:

$x^2h=7$

$h=\frac{7}{x^2}$

Surface area of the rectangular box is: $S=2lb+2lh+2hb$

Substitute l=x and b=x in above formula.

$S=2x^2+2xh+2xh$

$S=2x^2+4xh$

Now substitute $h=\frac{7}{x^2}$ in above equation.

$S=2x^2+4x(\frac{7}{x^2})$

$S=2x^2+\frac{28}{x}$

Now differentiate the surface area with respect to x.

$S'=4x-\frac{28}{x^2}$

Substitute S'=0 in above

$4x-\frac{28}{x^2}=0$

$\frac{28}{x^2}=4x$

$28=4x^3$

$7=x^3$

$x=\sqrt[3]{7}$

Substitute the value of x in $h=\frac{7}{x^2}$

$h=\frac{7}{(\sqrt[3]{7})^2}$

$h=\sqrt[3]{7}$

Hence, the dimensions are  $x=\sqrt[3]{7}$ and $h=\sqrt[3]{7}$.