The following sketch shows how to make an open-top box from a flat sheet of cardboard with cuts (heavy lines) and folds (dashed lines) as indicated. The dimensions of the resulting box depend on the length x of the corner cuts. 12 40 20 a. Show ow the volume of the box depends on length x of the corner cuts.

i. Excess each dimension of the box-length, width, and height-as a furation of x.

ii. Us: the formula for volume V of a box to write V as a function of x.

iii. Write the volume expression in expanded standard polynomial form.

b. Analyze the volume function for the box by answering the following questions.

i. Sketch a graph of y = V(x) for values of x that make sense in this situation.

ii. Estimate coordinates of the x-intercepts and local maximum and minimum points on the graph of V(x).

iii. Explain what the x-intercepts and local maximum and minimum points tell about the way the volume of such a box depends on the length x of the cuts.