Question

A thin closed rectangular box is to have one edge equal to twice the other end a constant volume 72 metre cube.find the least surface area of the box ​

Answer 1

Answer:

108 m²

Step-by-step explanation:

Let say one side = x

Then other Side = 2x

Third Side = Volume/ ( x * 2x)

=> Third Side = 72/ (2x² )

=>  Third Side = 36/x²

Surface Area =  2( x * 2x  + x *(36/x²)  + 2x*(36/x²)

= 2 * ( 2x²  + 36/x + 72/x)

= 2 * ( 2x²  + 108/x)

= 4x² + 216/x

dS/dx  =  8x  - 216/x²

8x  - 216/x² = 0

=> x³ = 27

=> x = 3

d²S/dx²  =  8 + 512/x³  > 0  hence least surface area for x = 3

Least surface Area = 4(3)² + 216/3  = 36 + 72 = 108 m²

x = 3  , 2x = 6

Third side = 72/(3 * 6) = 4

Surface Area = 2 ( 3 * 4 + 3*6  + 4 * 6)  = 2 * 54 = 108 m²