A cardboard box without a lid is to have a volume of 19,652 cm3. Find the dimensions that minimize the amount of cardboard used. (Let x, y, and z be the dimensions of the cardboard box.)

The dimension of the cardboard is 34 cm by 34 cm by 17 cm.

Step-by-step explanation:

Let the dimension of the cardboard box be x cm by y cm by z cm.

The surface area of the cardboard box without lid is

f(x,y,z)= xy+2xz+2yz.....(1)

Given that the volume of the cardboard is 19,652 cm³.

Therefore xyz =19,652

......(2)

putting the value of z in the equation (1)

The partial derivatives are

To find the dimension of the box set the partial derivatives and .Therefore

.......(3)

and  

.......(4)

Now putting the x in equation (3)

⇒y=34 cm

Then =34 cm.

Putting the value of x and y in the equation (2)

  =17 cm.

The dimension of the cardboard is 34 cm by 34 cm by 17 cm.