Question 1172105
Hi there,
Total Surface Area = 6x^2 (top and bottom) + 8xh (4 sides) = 200 cm^2
Tot SA = 6x^2 + 8xh = 200
Finding 'x' value of 'h'
8xh = 200 - 6x^2
h = (200 - 6x^2)/8x
Cancelling 2 top and bottom
h = (100 - 3x^2)/4x

Volume = length * breadth * height   (Using * for multiply)
V= 3x * x * h
V = 3x^2h
Bringing in 'x' value for 'h'
V(x) = 3x^2(100 - 3x^2)/4x
3/4x (100 - 3x^2)
V(x) = (300/4)x - 9x^3/4
Cancel down
V(x) = 75x - 9x^3/4
Differentiate
V'(x) = 75 - 27x^2/4
Make V'(x) = 0
75 - 27x^2/4 = 0
-27x^2/4 = - 75
Multiply both sides by -1
27x^2/4 = 75
27x^2 = 300
x^2 = 300/27 or 100/9 
x = 10/3.

.........................................................  -    10/3    +
Nature Table.75 - 27x^2/4 ...... +     0      -
 ........................................................... /     -      \         
Maximum
height = (100 - 3x^2)/4x
Height = [100 - 3(10/3)^2]  / 4 (10/3)
Height = 5 cm
Total Volume = length * breadth * height
Total volume = 3(10/3) * (10/3) * 5
Total Volume = 10 * 10/3 * 5
Total volume = 166.66 cm^3   
Hope this helps :-)