Volume of a Box A box is constructed by cutting
out square corners of a rectangular piece of cardboard
and folding up the sides. If the cutout corners have
sides with length x, then the volume of the box is given by the polynomial
A box is constructed from a square piece of metal that
is 20 inches on a side.
(a) If the square corners of length x are cut out, write
a polynomial that gives the volume of the box.
(b) Evaluate the polynomial when x=4 inches.
(c) Factor out the greatest common factor for this
polynomial expression
well, the best I can see is the base area is (20-2x)^2, height is x
volume=base area*height
(a) To find the volume of the box, we need to determine the dimensions of the box after the square corners are cut out.
Given that the original square piece of metal is 20 inches on each side, and the cutout corners have sides with length x, the length and width of the rectangular piece of cardboard that remains after the corners are cut out will be (20 - 2x) inches.
The height of the box will be equal to the length of the cutout corners, which is x inches. Therefore, the volume of the box can be calculated by multiplying the length, width, and height:
Volume = (20 - 2x) * (20 - 2x) * x
To simplify this polynomial expression, we can expand it:
Volume = x * (400 - 40x - 40x + 4x^2)
Volume = x * (400 - 80x + 4x^2)
Volume = 4x^3 - 80x^2 + 400x
So, the polynomial that gives the volume of the box is 4x^3 - 80x^2 + 400x.
(b) To evaluate the polynomial when x = 4 inches, we substitute x = 4 into the expression:
Volume = 4(4)^3 - 80(4)^2 + 400(4)
Volume = 4(64) - 80(16) + 400(4)
Volume = 256 - 1280 + 1600
Volume = 576 cubic inches
Therefore, when x = 4 inches, the volume of the box is 576 cubic inches.
(c) To factor out the greatest common factor for the polynomial expression 4x^3 - 80x^2 + 400x, we can identify the highest power of x that every term has in common, which is x.
Factoring out x, we get:
4x(x^2 - 20x + 100)
Therefore, the greatest common factor for the polynomial expression is 4x, and the factored form is 4x(x^2 - 20x + 100).