Answer to Question #20347 in Calculus for Sara

Question

A cube has a total volume of 120 m^3. Its length must be three times as long as its width. Find the dimensions of such a cube with the lowest surface area.

Answer 1

Surface area S=2*(a*b+b*c+a*c)=(from the taska=3b)=2*(9b^2+4b*c)
V=a*b*c=120, so c=120/(a*b)=40/b^2
So, S=2*(9b^2+160/b)
It's derivative is:s'=36b-160/b^2
Minimum is at point s'=0
36b=320/b^2
B^3=320/36=80/9=> b=2SQRT3(10/9)
a=3b=6sqrt3(80/9)
c=120/(a*b)=120/12sqrt3(10/9)^2=10*sqrt3(81/100)=30*sqrt3(0.03)