A cardboard box without a lid is to have a volume of 23,328 cm3. Find the dimensions that minimize the amount of cardboard used. (Let x, y, and z be the dimensions of the cardboard box.)

x,y,z=36cm, 36cm, 18cm

Step-by-step explanation:

Volume of the Cardboard =

The dimensions of the box are x, y and z.

Volume of a Cuboid = xyz.

Since the top is open, there is only one 'xy' side.

Surface Area, f(x,y,z)= xy+2yz+2xz

We take the first derivative of f(x,y)

Equate the derivatives to zero and solve for x and y respectively.

Substituting x into y and vice versa

Recall:

x,y,z=36cm, 36cm, 18cm

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is