Question

Squares of side x are cut from each corner of an 8 in x 5 in rectangle (see figure), so that its sides can
be folded to make a box with no top Define a function in terms of x that can represent the volume of this
box
8 inches
5 inches
they will be​

Answer 1

Answer:7. A person is going to make a box by taking a square piece of cardboard, which is 12 inches on a side, like the one in the picture to the right, cutting squares out of the corners, and folding the edges like the other figure in the picture to the right. How big of a square should they cut out of the corners in order to maximize the volume of the resulting box?

This is a classical problem which has delighted calculus students down through the ages.

We first express the volume of the box as a function of x, the side of the square which is cut out of each corner.

The resulting box will have a base which is a square which all of whose sides will have length 12 - 2x. The height of the box will be x. Therefore, the volume will be

V = (12 - 2x)2x

= 144x - 48x2 + 4x3

Answer 2

Given:

An 8 in x 5 in rectangle

A square of side x (used to cut the corners of the rectangle)

Required:

Function in terms of x that can represent the volume if this box

Solution:

Let’s draw the given figure to visualize what is being asked.

Let a be the width of the rectangle

Let b be length of the rectangle

After we cut the corners and fold the sides forming the topless box. The rectangle is now a hollow rectangular parallelepiped (box) with height x.

The formula to find the Volume of a rectangular parallelepiped is:

V= lwh

Where:

l= length of the solid

w= width of the solid

h= height or thickness

Substitute the equation obtained from the figure,

V= a•b•x

V= (5 - 2x) (8 - 2x) x

= (40 - 26x + 4x^2) x

The final answer is

V(x)= 40x - 26x^2 + 4x^3

You’re welcome❤️