We want to construct a closed rectangular box whose base has a length three times the width. The surface area has to be 600 cm2. Determine the dimensions of the box that will produce the largest volume.

Answer 1

L,w,h

area = 2 Lw + 2 Lh + 2wh = 600
so
Lw+Lh+wh = 300
but L = 3w
so
3 w^2 + 4 w h = 300
now
V = L w h = 3 w^2 h
so h = V/3w^2

3 w^2 + 4 w(V/3w^2) = 300

4 V/3w = 300 - 3 w^2

(4/3)V = 300 w - 3 w^3

we want d(4/3)V /dw = 0

300 = 9 w^2
w = (10/3)sqrt 3
L = 10 sqrt 3

w L = 100
so h = 6
check my arithmetic!

Answer 2

Thank you!!!

Answer 3

You are welcome.