Using the Minimization of Area,
The dimensions of rectangular box are 20 and 40.
We have given that,
Volume of rectangular box = 32,000cm³
Assume b is the side of square base and h is the height of the box.
The formula to find the volume is equal to product of area of base and height
V = b²h
=> h = V/b² --(1)
The formula for the surface area is
A = b×b + 2b×h +2b×h = b² + 4bh
using equation (1) we get,
=> A = b² + 4V/b
differentiating Area (A) with respect to b
to find the minima and putting dV/db = 0
dA/db = 2b - 4V/b²
=> 2b - 4V/b² = 0
=> b³ = 4V/2
Plunging the value of volume in above equation we get,
b³ = 4 (32000)/2
=> b = 40 and
h = 32000/ 40×40 = 20
Thus , the dimensions of the rectangular box are 40 and 20.
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