An open-top box is to be constructed from a 6 foot by 8 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out.
1. Find the function V that represents the volume of the box in terms of x
2.How do I graph this gfunction to show the valid range of the variable x.
One side of the box is 6 - 2x and the other is 8 - 2x. When folded up, the height of the side of the box will be x.
V = x (8-2x)(6-2x)
The range of x is from 0 up to 3. When x - 3, one of the box side lengths becomes zero, and so does the volume.
Make your graph of V vs x in the usual way.
To find the function V that represents the volume of the box in terms of x, we need to determine the dimensions of the box after cutting out the squares and folding up the flaps.
Since one side of the box is 6 feet and the squares are cut from each corner, the remaining length of that side would be (6 - 2x) feet. Similarly, the remaining length of the other side would be (8 - 2x) feet.
The height of the box is equal to the length of each side of the square that was cut out, which is x feet.
Thus, the volume V of the box can be represented as the product of the three dimensions:
V = x(8 - 2x)(6 - 2x)
To graph this function and show the valid range of the variable x, follow these steps:
1. Determine the valid range of x:
The range of x is from 0 up to 3. When x = 3, one of the box side lengths becomes zero, resulting in a volume of zero as well.
2. Create a graph with the x-axis representing the values of x and the y-axis representing the volume V.
3. Plot points on the graph by substituting different values of x into the equation V = x(8 - 2x)(6 - 2x). For example, you can choose values such as x = 0, 1, 2, and 3.
4. Connect the plotted points with a smooth curve.
5. Shade the area of the graph where the values of x are between 0 and 3, indicating the valid range. Ensure that the shaded area does not extend beyond x = 3.
Note: This graph will show how the volume of the box changes as the side length of the square cut-out (x) varies within the valid range.