Final answer:
The distance traveled by a particle with position (x, y) as t varies in the given time interval can be found by using the distance formula and integrating the distance function.
Step-by-step explanation:
The position of a particle can be found by substituting the given equations for x and y into the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, x₁ = 2sin²(t), y₁ = 2cos²(t), x₂ = 2sin²(t'), and y₂ = 2cos²(t').
Substituting these values into the distance formula and simplifying, we get:
d = √(8sin²(t') - 8sin²(t) + 8cos²(t') - 8cos²(t))
To find the total distance traveled by the particle in the given time interval, we need to find the definite integral of the distance function:
D = ∫₀⁵ √(8sin²(t') - 8sin²(t) + 8cos²(t') - 8cos²(t)) dt'