Final answer:
To maximize the area of each windmill blade while adhering to a 10-meter perimeter constraint, the isosceles triangular blades should be designed with the two equal sides as long as possible to increase the height and therefore the area, while the rest of the perimeter forms the base.
Step-by-step explanation:
To maximize the area of the windmill blades while ensuring that the total perimeter is no more than 10 meters, we must consider an isosceles triangle. The isosceles triangle will have two sides of equal length (the blades) and a base. To find the optimal dimensions, we use the formula for the area of a triangle (Area = 1/2 × base × height) and the perimeter (Perimeter = base + 2×side).
Under the constraint that the perimeter must be less than or equal to 10 meters, we must allocate this length between the base and the two sides. Keep in mind that a larger height will yield a greater area, which suggests that the base should be shorter to allow more length for the sides, subsequently increasing the height of the triangle. By applying calculus and considering the triangle's properties, the maximum area can be achieved when the two equal sides are as long as possible, with the remainder of the perimeter forming the base.
A practical approach for this design problem is to use optimization techniques to derive the specific lengths that maximize the triangular blade's area under the given perimeter constraint. In reality, engineering considerations such as material strength and airflow dynamics would also play a role, beyond the simplistic mathematical model.