A160-inch strip of metal 20 inches wide is to be made into a small open trough by bending up two sides on the long side, at right angles to the base. the sides will be the same height, x. if the trough is to have a maximum volume, how many inches should be turned up on each side?

The value of double derivative at x=4.834 is negative, therefore the trough have a maximum volume at x=4.834 inches.

Step-by-step explanation:

The dimensions of given metal strip are

Length = 160 inch

Width = 20 inch

Let the side bend x inch from each sides to make a open box.

Dimensions of the box are

Length = 160-2x inch

Breadth = 20-2x inch

Height = x inch

The volume of a cuboid is

Volume of box is

Differentiate with respect to x.

Equate V'(x)=0, to find the critical points.

Using quadratic formula,

The critical values are

Differentiate V'(x) with respect to x.

The value of double derivative at critical points are

Since the value of double derivative at x=4.834 is negative, therefore the trough have a maximum volume at x=4.834 inches.

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is