a box with no top is to be constructed from a piece of cardboard with sides of length A and B by cutting out squares of length h from the corners and folding up the sides find the value of h that maximizes the volukme of the box if A=17 and B=27

The value of h that maximizes the volume of the box is h = 3.36

Step-by-step explanation:

First, the volume of a box of length L, width W, and height H is:

V = L*W*H.

in this case, we start with a rectangle of measures AxB.

We can define that the side with length A will be the length, and the other side will be the width.

As we cut squares of length h in the corners, now we will have that the measures of the box are:

Length = A - 2*h

Width = B - 2*h

Height = h.

And we know that A = 17 and B = 27.

So we can replace those values and get:

Length = 17 - 2*h

Width = 27 - 2*h

Height = h

The volume of this box will be:

V = (17 - 2*h)*(27 - 2*h)*h = 459*h -54*h^2 - 34*h^2 + 4*h^3.

Now we want to maximize this.

V(h) = 4*h^3 - 88*h^2 + 459*h

To find the maximum, let's look at the first derivation of V(h)

V'(h) = 3*4*h^2 - 2*88*h + 459.

We need to look for which values of h this is zero.

0 = 12*h^2 - 176*h + 459

The solutions are given by the Bhaskara equation:

Then the two solutions are:

h = (176 + 95.33)/24 = 11.3

h = (176 -95.33)/24 = 3.36

Now, the problem with the first solution is that if we cut two squares of length 11.3 in side A (2*11.3 > 17) , the amount of material cut is larger than A, so that option can be discarded.

Then the only option is h = 3.36

The value of h that maximizes the volume of the box is h = 3.36

answer:

it is general formula that is used to write value to its excet position

a11 a12 a13

a21 a22 a23

it is the example of order 2 by 3 matrix in this

a11 means value occur in 1 st row 1st column

a12 means value occurs in 1st row 2nd column

a13 means value occur in 1st row and 3rd column

a21 means value occurs in 2nd row and 1st column

a22 means value occurs in 2nd row and in 2nd column

a23 means value occurs in 2nd row and in 3rd column