Expert: Alon Mandes - 11/24/2008

Question
A rectangular storage container with a closed top is to have a volume of 24 meters cubed. the length of its base is twice the width. Material for the top and the base cost $9 per square meter.material for the sides costs $8 per square meter. Find the cost of materials for the cheapest such container.
please help me answer if u can!!!

Answer
To find the cost of materials for the cheapest container, we must
1st find the dimensions of such container. We know so far that :
1. Length of the base is twice the width. So if the width is x
then the length will be 2x. & the volume will be x*2x*h. "where
h is the height".
2. The volume is 24 -> 2hx^2=24 -> h=12/x^2.
Now, we want to calculate , the coast of the materials :
9$ for top & base -> coast=x*2x*9$+x*2x*9$=36x^2 $ .
8$ for sides -> coast=2(h*x)8$+2(h*2x)8$=2(12/x)8$+2(24/x)8$=
192/x + 384/x $ .
Total coast T(x)=192/x + 384/x + 36x^2. Let's minimize this
function : T(x)=576/x+36x^2 -> T'(x)=-576/x^2 + 72x. Let's set
T'(x)=0 -> 72x=576/x^2 -> x^3=8 -> x=2. That means that the dimensions of the container will be :
Length = 4
Width = 2
Height = 12/4=3
& the minimal coast will be : 576/2+36*2^2=288+144=432 $.

Alon.