Question

A rectangular tank with a square​ base, an open​ top, and a volume of 1372 ft cubed is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

Answer 1

let length of tank is x

width of the tank is x

height of the tank is y

Volume of the rectangular box

V=length*width*height

V=x^2y

1372=x^2y

y=1372/x^2 ------------(1)

surface area of the tank

(putting value of y from equation (1))

---------------------(2)

Differentiating with respect to x

(Because )

For critical points

second derivative

at x=14

so x=14 ft is minima point

from equation (1)

y=1372/(14)^2=7 ft

length x=14 ft

width x=14 ft

height y = 7 ft