Question

A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.(a) Sides of base (cm)(b) Height (cm)

Answer 1

Answer:

Step-by-step explanation:

Given

Volume of box $V=32,000 cm^3$

Let b be the square base and h be the height of box

Volume $V=b^2\cdot h$

$h=\frac{V}{b^2}$

Surface Area $A=b^2+4bh$

$A=b^2+4b\times \frac{V}{b^2}$

$A=b^2+4\frac{V}{b}$

Differentiate A w.r.t b to get maxima/minima

$\frac{\mathrm{d} A}{\mathrm{d} b}=2b-\frac{4\times V}{b^2}$

$2b-\frac{4\times V}{b^2}=0$

$b^3=\frac{4V}{2}$

$b^3=\frac{4\times 32,000}{2}$

$b=40 cm$

$[tex]h=\frac{32,000}{40\times 40}$

$h=20 cm$