Web Analytics
Answered

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.

A14.12cm

B16.14cm

C12.13cm

D10.36cm

Verified Answer :

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 \(h=\sqrt[3]{\frac{16,823}{4}} =16.14 \text{ cm}\)

Related Questions