A rectangular block is constructed so that its length is 3 times the width. Find the dimension of the rectangular block that will give the least possible surface area if the volume of the block is 81 cubic meters.

Answer 1

The block's dimensions are w, 3w, h
v = 3w^2h = 81, so h = 81/(3w^2) = 27/w^2
The area is
a = 2(w*3w + wh + 3wh) = 6w^2 + 4wh = 6w^2 + 4w*27/w^2 = 6w^2 + 108/w
for minimum area, you want da/dw = 0, so
da/dw = 12w - 108/w^2 = 12(w^3-9)/w^2
da/dw=0 when w^3-9 = 0
w = ∛9
...

Answer 2

a square cross-section gives the least surface area

so the volume is ... 81 = 3 w * w * w = 3 w^3

the block is 3 x 3 x 9

to Steve ... you lost a factor of two in your area calculation ... a = ....

Answer 3

rats. Way to watch.

Answer 4

Im so confused with these. Can anyone elaborate more on this?