A box with an open top is to be made by cutting 5-inch squares from the corners of a rectangular piece of cardboard whose length is twice its width and then folding up the remaining flaps. Let x represent the width of the original piece of cardboard; express the volume (V) of the box as a function of x, and simplify the function.
Width = x inches (given)
Length = 2x inches(given)
Height = 5 inches (given)
Volume = 5*x*2x = 10x² in.3
To find the volume of the box, we need to understand its dimensions.
Let's start by visualizing the rectangular piece of cardboard. We're given that its length is twice its width. So, we can say that the length of the cardboard is 2x, where x represents the width. The width is x, and the height, when the flaps are folded up, will be 5 inches.
To create the box, 5-inch squares are cut from each corner, which means that the width and length of the base of the box will decrease by 10 inches (5 inches on each side).
So, after cutting the squares, the dimensions of the base of the box will be (2x - 10) inches by (x - 10) inches. The height of the box will be 5 inches.
Now, we can calculate the volume of the box:
V = Length * Width * Height
= (2x - 10) * (x - 10) * 5
= 5(x - 10)(2x - 10)
= 5(2x^2 - 30x - 20x + 100)
= 5(2x^2 - 50x + 100)
= 10x^2 - 250x + 500
Therefore, the volume of the box as a function of x is V(x) = 10x^2 - 250x + 500.