a rectangular piece of cardboard measuring

16

in by

24

in is to be made into a box by cutting equal size squares from each corner and folding up the sides. let x represent the length of a side of each such square in inches. complete parts (a) through (d) below.

(a) give the restrictions on x.

(b) determine a function v that gives the volume of the box as a function of x.

(round to the nearest hundredth.)

(c) for what value of x will the volume be a maximum?

(round to the nearest hundredth.)

(d)the volume will be greater than

288 cubic inches when

(type an integer or a decimal. round to the nearest thousandth.)

(a) x<8

(b)

(c)

(d)

Step-by-step explanation:

Optimization

It is the procedure to find the set of values for the variables of a function such that it reaches a maximum or a minimum value. If equalities are given as relationships between the variables, then the derivative is a suitable method to find the critical points or candidates for extrema values.

The problem at hand is about a geometric maximization, given some dimensional conditions. First, we have a rectangular piece of cardboard measuring 24 x 16 inches. A box is to be made out of that cardboard by cutting equal size squares from each corner and folding up the sides of length x.  

(a)

When we do so, the base of the box will have dimensions

Since the new width of the base must be positive, then

which poses the restriction

The same situation happens with the length

x<8

Since this last condition is more restrictive than the first, we state that x must be less than 8

(b) The volume of the box is the product of the area of the base by the height x

Operating

(c)

To find the maximum volume, we take the first derivative of V:

Equating to zero to find the critical points

Dividing by 4:

The roots of the equation are:

Since x=10.19 is out of the restrictions found in part a, the only valid solution is

We must test if the critical point is a maximum or a minimum, by computing the second derivative

Since the second derivative is negative, the value is a maximum

(d) We must find all the values of x such as v>288:

Rearranging

Simplifying by 4

Factoring

We found three real and positive roots for the third-degree polynomial

The function is positive when

or

The only interval lying into the valid values of x is