A wooden box without a top cover is to be made from layers of plywood (see figure below). The base is designed to be three layers thick: Maths Assignment, NTU, Singapore

Question 1

A wooden box without a top cover is to be made from layers of plywood (see figure below). The base is designed to be three layers thick, the sides to be two layers thick and the front and back to be one layer thick.

• Formulate a function giving the total area of plywood used that depends on the dimensions (x, y, z) of the box.
• If the volume of the box is required to be 3 m3, state the constrained optimization problem for the box to minimize its plywood usage and solve it for the desired dimensions.
• Without using any calculations, explain logically why the dimensions solved in Question 1(b) result in a constrained minimum (but not maximum) plywood usage.

Question 2

The pressure distribution on a functional rectangular steel plate supporting a load is modeled by the:

where x and y are in meters.

• Calculate the gradient of the pressure function. Hence, determine the rate of change of pressure at the point (0, 1) in the direction of u = i – 2j.
• If the dimensions of the plate are not known, determine the location on the plate where the pressure is maximum and calculate this pressure. Show that this pressure is both a local and global maximum.
• If the plate is 1 m long by 2 m wide, such that 0 ≤x≤ 1, 0 ≤ y ≤2, determine the minimum pressure on the plate. State any theorems used if any.

Question 3

• Determine the domain and range of the function below and sketch the domain (shaded) on the xy-plane. Describe its level set.

• Determine the domain and range of the function below. Describe its level set.