Suppose you take a piece of cardboard measuring 7 inches by 7 inches, cut out square corners with sides x inches long, and then fold up the cardboard to make an open box. Express the volume V of the box as a function of x.
Yes so my conclusion is that
V(x) = x (7 – 2x)^2
is this correct?
did you make a sketch?
isn't the base after the cut-out (7-2x) by (7-2x) and isn't the height of the box x inches?
volume of box = base area x height
=
correct
Thanks! I knew I would get this stuff! Thanks for your help!
To express the volume V of the box as a function of x, we need to compute the volume of the open box.
Step 1: Determine the dimensions of the box after the corners are cut out.
When the square corners are cut out, the length and width of the cardboard decrease by 2x inches (since there are two corners per side). Therefore, the length of the box would be (7 - 2x) inches, and the width of the box would also be (7 - 2x) inches.
Step 2: Calculate the height of the box.
The height of the box would be the height of the original cardboard, which is 7 inches.
Step 3: Compute the volume of the box.
The volume (V) of a rectangular box is calculated by multiplying its length (l), width (w), and height (h) together. In this case, the volume can be expressed as:
V = length * width * height
= (7 - 2x) * (7 - 2x) * 7
= 49x^2 - 98x^2 + 49x
Therefore, the volume (V) of the box can be expressed as a function of x: V(x) = 49x^2 - 98x^2 + 49x.