Question

A Rectangular box with square base and open at the top is to have a capacity of 16,823 cu.cm. Find the height of the box that requires minimum amount of the material required.
a. 14.12cm

b. 16.14cm

c. 12.13cm

d. 10.36cm

Answer 1

let the side of base =x and height=h
 
$hx^2=16,823$ so $x=\sqrt{\frac{16,823}{h}}$
 
surface area of the box = S(x)= x² +4xh
 
to find the minimum we calculate the first derivative
S'(x) = 2x + 4h
this is minimum for S'(x) =0
 
2x + 4h = 0 replace x by its value..
$2\sqrt{\frac{16,823}{h}}+4h=0$
$4h=-2\sqrt{ \frac{16,823}{h} }$ square both sides
 
$16h^2=4\frac{16,823}{h}$ cross multiply
 
$h^3=4\frac{16,823}{16}$
 
so $h=\sqrt[3]{\frac{16,823}{4}} =16.14 \text{ cm}$