An open box is to be made from a flat piece of material 18 inches long and 5 inches wide by cutting equal squares of length xfrom the corners and folding up the sides.
Write the volume Vof the box as a function of x. Leave it as a product of factors, do not multiply out the factors.
V=
If we write the domain of the box as an open interval in the form (a,b), then what is a=?
a=
and what is b=?
b=
To find the volume of the box, we need to determine the dimensions of the box in terms of the length x.
First, let's determine the dimensions of the base of the box:
Length of base = 18 inches - 2x (since two squares, one from each end, are removed)
Width of base = 5 inches - 2x (same reason as above)
The height of the box is simply the length of the squares, x.
Therefore, the volume V of the box can be expressed as:
V = (18 - 2x)(5 - 2x)(x)
To calculate the domain of the box, we need to consider the restrictions on the value of x. Since we cannot have negative dimensions or a base with zero area, we have the following inequalities:
18 - 2x > 0 (to ensure that the length of the base is positive)
5 - 2x > 0 (to ensure that the width of the base is positive)
Simplifying these inequalities:
18 > 2x
x < 9
5 > 2x
x < 2.5
Therefore, the domain of the box can be represented as the open interval (a, b), with a = 0 (since the value of x cannot be negative) and b = 2.5 (the upper limit of x that satisfies the inequalities).
So, a = 0 and b = 2.5.
To find the volume of the box as a function of x, we need to determine the dimensions of the box after folding.
First, let's consider the length of the box. We have an 18-inch long piece of material, and we will be cutting squares of length x from each corner. Since we will be folding up the sides, the final length of the box will be (18 - 2x) inches.
Next, let's consider the width of the box. We have a 5-inch wide piece of material, and we will be cutting squares of length x from each corner. Similar to the length, the final width of the box will be (5 - 2x) inches.
Finally, let's consider the height of the box. The squares that we cut from each corner will form the base of the box, so the height will be equal to x inches.
Now, for the volume of the box, we multiply the length, width, and height:
V = (length) * (width) * (height)
V = (18 - 2x) * (5 - 2x) * x
So, the volume of the box V, as a function of x, is given by:
V = x(18 - 2x)(5 - 2x)
To find the domain of the box as an open interval, we consider the values of x that make sense for the dimensions. Since we are cutting squares from the corners, x cannot exceed half the length or half the width of the original material. In other words, x ≤ 9 and x ≤ 2.5. Therefore, the domain would be (0, 2.5].
Therefore, a = 0 and b = 2.5.