Answer:
[tex]V(x)=(11-2x)(12-2x)x= 4 x^3- 46 x^2 + 132 x.[/tex]
Step-by-step explanation:
Consider the picture attached, which illustrates the process of making the box.
Observe that in figure (2) the sides of the rectangle were both reducing two times the length of the side of the square. Therefore, the sides of the new figure are 12-2x units and 11-2x units, respectively.
In figure (3), when the box is done, we see that its dimensions are 11-2x, 12-2x and x (width, length and high, respectively). Thus, by the formula of the volume of a box, we know that the volume V(x) is given by [tex]V(x)= width\times length\times high=(11-2x)(12-2x)x.[/tex] Solving the parenthesis, we get that [tex]V(x)=4 x^3- 46 x^2 + 132 x.[/tex]
The volume V of the box as a polynomial function in terms of x is;
V(x) = 4x³ - 46x² + 132x
We are told that the rectangle has dimensions as;
Length; L = 12 units
Width; W = 11 units
Now, we are told that squares of x by x units are cut out of each corner.
This means that the length will now be;
L_new = 12 - (x + x)
L_new = 12 - 2x
Similarly, for the width, we will have;
W_new = 11 - (x + x)
W_new = 11 - 2x
Now, it is pertinent to not that the height of the box will now be x as the dimension of the square will be the height since it is an open box. Thus;
Height; H_new = x
Formula for volume of a box is given as;
Volume = Length x Width x Height
Thus;
V(x) = (12 - 2x) * (11 - 2x) * x
Expanding this gives us;
V(x) = 4x³ - 46x² + 132x
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