This assignment has four problems, of equal weight. For each of the first three problems, write an algebraic linear program as we did in class.

Use the "standard" format, starting with clear definitions of the decision variables as quantities. For example, write "x₁ = number of regular widgets to produce," not "x₁ = regular". Once you have defined variables, write the problem in the form:

Customarily, the constraints involving more than one decision variable are listed first, followed by the simpler constraints like "x₁ >  0"

1   Text, p. 57, Exercise 2.

2   Text, p. 57, Exercise 4.

3   Text, p. 57, Exercise 6.

4   This problem has three parts.

• In R² consider:
  • the square of side 2 centered at the origin, i.e. the square with four vertices (-1,-1), (-1,1), (1,-1) and (1,1),
  • four circles of radius 1/2 inscribed in the corners of the square (i.e. circles of radius 1/2 centered at (-1/2,-1/2), (-1/2,1/2), (1/2,-1/2) and (1/2,1/2)), and
  • the smallest circle, with center at the origin, that is tangent to the above four circles.
  Calculate the radius of the inner circle.
• Repeat for R³. Now we have a cube of side 2, with 8 vertices, 8 balls of radius 1/2 at the 8 corners of the cube and an inner ball tangent to them. Calculate the radius of the inner ball.
• Repeat for Rⁿ, for n > 3. Conclude that the inner ball will eventually stick out of the cube. What is the smallest n for which this will happen?