Question

find the dimension of the rectanglar box without a top of maximum capacity whose surface area is 108 sq.cm.

Answer 1

Answer:

Dimension of Box = 6 * 6 * 3

Step-by-step explanation:

For a given Surface Area Volume is maximum

=> for a given Volume , Surface Area to be minimum

Let say Max Volume = V cm³

then Surface least area  = 108 cm²

Surface area = xy + 2(x + y) h

Volume = xyh  = V

=> h = V/xy

=> Surface Area = xy + 2V(x + y) /xy

=> S = xy  + 2V/y  + 2V/x

δS/δx  = y  - 2V/x²  => x²y = 2V

δS/δy = x - 2V/y²    => xy² = 2V

=>  x²y =  xy²

=> x = y  

δ²S/δx² = 4V/x³  & δ²S/δy² = 4V/y³  is +ve

Hence for x = y , surface area is least

& x³ = y³ = 2V      

also xyh = V  

=> x²h = V

=> x/h = 2

=> x = 2h

Surface Area =

xy + 2(x + y) h = 108

=> 2h * 2h  + 2(2h + 2h)h = 108

=> 4h² + 8h² = 108

=> 12h² = 108

=> h² = 9

=> h = 3

x = y = 6

Dimension of Box = 6 * 6 * 3