Final answer:
The dimensions that produce minimum and maximum areas of an equilateral triangle and a square with a sum of perimeters of 10 depend on the allocation of side lengths. For minimum area, an optimization method is needed. For maximum area, it involves setting one shape's side length to nearly zero.
Step-by-step explanation:
The solution to this optimization problem involves calculating the minimum and maximum combined areas of the equilateral triangle and square given a fixed perimeter sum.
Part (a): To minimize the area, set the side length of the triangle t and the square s. The perimeters are 3t for the triangle and 4s for the square, with 3t + 4s = 10 as the equation. To minimize the combined area, you could use the method of Lagrange multipliers or differentiate the total area with respect to one variable after expressing the second variable in terms of the first one from the perimeter constraint equation.
Part (b): For maximizing the area, since both shapes are regular, the situation would correspond to one of the figures being set with a minimum side length (approaching zero), giving nearly all the perimeter to the other figure. In real-world practical terms for this question, there is no upper bound for area, as one shape's area can increase indefinitely while the other's approaches zero.
Dimension analysis is critical in ensuring the proper interrelationship between perimeter and area: perimeter is linear and has units of length, while area is quadratic with units of square length. The provided formulas confirm dimensional consistency: for a square with side length a, perimeter is 4a, and area is a². For a circle inscribed in that square, we could estimate the circle's perimeter as close to 3a, given its diameter is a.