VCE Methods Differential Calculus Application Task 6

Question 1 [2021 Exam 2 Section B Q1]

A rectangular sheet of cardboard has a width of \(h\) centimetres. Its length is twice its width. Squares of side length \(x\) centimetres, where \(x > 0\), are cut from each of the corners, as shown in the diagram below.

The sides of this sheet of cardboard are then folded up to make a rectangular box with an open top, as shown in the diagram below. Assume that the thickness of the cardboard is negligible and that \(V_{box} > 0\).

A box is to be made from a sheet of cardboard with \(h = 25\) cm.

a. Show that the volume, \(V_{box}\), in cubic centimetres, is given by \(V_{box}(x) = 2x(25 – 2x)(25 – x)\). 1 mark

b. State the domain of \(V_{box}\). 1 mark

c. Find the derivative of \(V_{box}\) with respect to \(x\). 1 mark

d. Calculate the maximum possible volume of the box and for which value of \(x\) this occurs. 3 marks

e. Waste minimisation is a goal when making cardboard boxes. Percentage wasted is based on the area of the sheet of cardboard that is cut out before the box is made. Find the percentage of the sheet of cardboard that is wasted when \(x = 5\). 2 marks

Now consider a box made from a rectangular sheet of cardboard where \(h > 0\) and the box's length is still twice its width.

f.

i. Let \(V_{box}\) be the function that gives the volume of the box. State the domain of \(V_{box}\) in terms of \(h\). 1 mark

ii. Find the maximum volume for any such rectangular box, \(V_{box}\), in terms of \(h\). 3 marks

g. Now consider making a box from a square sheet of cardboard with side lengths of \(h\) centimetres. Show that the maximum volume of the box occurs when \(x = \frac{h}{6}\). 2 marks