A machine produces open boxes using rectangular sheets of metal (12in by 20in). The machine cuts equal-sized squares from each corner, and then shapes the metal into an open box by turning up the sides.
Express the volume of the box V(x) in cubic inches as a function of the length of the side of the square curs
Find the domain of the function V(x)
V(x) = length*width*height = (12-2x)(20-2x)(x)
To find the volume of the box as a function of the side length of the square cut, we first need to determine the dimensions of the box.
Let's assume that the side length of the square cut is "x" inches.
When the square is cut from each corner of the rectangular sheet, the dimensions of the resulting box will be:
- Length = 20 inches - 2x (since a square is cut from both ends)
- Width = 12 inches - 2x (again, a square is cut from both ends)
- Height = x inches (this is the side length of the square cut)
To calculate the volume of the box, we multiply these dimensions together:
V(x) = (20 - 2x)(12 - 2x)(x)
Now let's find the domain of the function V(x). The domain refers to the values of x that make the function valid. Since we are dealing with lengths, the domain will be restricted to positive values of x.
To determine the valid range for x, we consider the dimensions of the box. For the box to have positive dimensions, the length and width must be greater than zero:
20 - 2x > 0 ---> 2x < 20 ---> x < 10
12 - 2x > 0 ---> 2x < 12 ---> x < 6
Hence, the domain of the function V(x) is 0 < x < 6. This means x can take any value between 0 and 6, but not including 0 or 6.
Therefore, the volume of the box V(x) is given by the function V(x) = (20 - 2x)(12 - 2x)(x), where x belongs to the interval (0, 6).