Volume of a Box (Refer to the preceding exercise.)
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 inches.
(c) Factor out the greatest common factor for this
polynomial expression.
what shape is the box? Is it a square box, of height x? Or is it a cube? Does it have a lid, or is it open?
To find the volume of the box, we need to determine the dimensions of the box after the square corners are cut out.
(a) The length of each side of the box will be reduced by 2x (x on each side) since square corners of length x are cut out. Therefore, the dimensions of the box will be (20 - 2x) inches by (20 - 2x) inches, and the height of the box will be x inches. The volume of a box is given by the formula: Volume = Length x Width x Height. So, in this case, the volume of the box can be represented by the polynomial expression:
V(x) = (20 - 2x)(20 - 2x)(x)
(b) To evaluate the expression V(x) when x = 3 inches, substitute x = 3 into the polynomial:
V(3) = (20 - 2(3))(20 - 2(3))(3)
= (20 - 6)(20 - 6)(3)
= (14)(14)(3)
= 588 cubic inches
Therefore, when x = 3 inches, the volume of the box is 588 cubic inches.
(c) To factor out the greatest common factor for the polynomial expression, we can observe that each term of the polynomial has a common factor of 2, so we can factor it out:
V(x) = 2 * (10 - x)(10 - x)(x)
Therefore, the greatest common factor for the polynomial expression is 2, and factoring it out gives the simplified form of V(x) = 2(10 - x)(10 - x)(x).