help me please explain?

Question

An open-top box is to be constructed from a 4 foot by 6 foot rectangular cardboard by cutting out equal squares at each corner and folding up the flaps. Let x denote the length of each side of the square to be cut out.
a) Find the function V that represents the volume of the box in terms of x.
Answer:
Function you are looking for is V = 4(3 - x)(2 - x)x = 4x(x -3)(x – 2)

b) Graph this function and show the graph over the valid range of the variable x..
Show Graph here.





c) Using the graph, what is the value of x that will produce the maximum volume?
Answer.

Answer 1

You must know there's no way to show a graph here.

V = x(4-2x)(6-2x)

max when derivative of V = 0
V = 4x^3 - 20x² + 24x
V' = 12x² - 40x + 24
0 = 12x² - 40x + 24
0 = 3x² - 10x + 6
x = 10/6 ±√100 - 72)/6
x = 5/3 ± √28 / 6
x = 5/3 + 2√7 / 6
x = (5 + √7)/3 = 2.549

Answer 2

The result of removing squares of side length x from each corner of the 4 foot sides is flaps that measure 4 - 2x.

The result of removing squares of side length x from each corner of the 6 foot sides is flaps that measure 6 - 2x.

The height of the open top box is x when the flaps are folded up to form the box.

Volume = (length)(width)(height)
V(x) = (4 - 2x)(6 - 2x)x
factor a -2 from each binomial
V(x) = (-2)(-2 + x)(-2)(-3 + x)x
write each binomial in standard form
V(x) = 4(x - 2)(x - 3)x
V(x) = 4x(x - 3)(x - 2)

The shortest side of the original rectangle is 4 feet, so the side length of the squares (that are removed from each corner) range between 0 and 2 feet. That is, 0 < x < 2, which is the valid range of x. At, x = 0 and x = 2, we can not fold a box from the cardboard.

Create a table of coordinates of points on the graph of V(x).
(x, V(x))
-----------
(0, 0)
(0.25, 4.8125)
(0.5, 7.5)
(0.75, 8.4375)
(1, 8)
(1.25, 6.5625)
(1.5, 4.5)
(1.75, 2.1875)
(2,0)

Plot these points and connect with a smooth curve to graph V(x). Place "open circles" at the points (0, 0) and (2, 0).

Using this graph, x = 0.75 is the approximate value of x that will produce the maximum value of the box.

Answer 3

Do your own homework.

Answer 4

you should listen to your teacher