Final answer:
The displacement of the particle is -2301 feet and the total distance traveled is 2301 feet.
Step-by-step explanation:
(a) Displacement is calculated by integrating the velocity function. To find the displacement, we integrate the velocity function over the given interval (1 ≤ t ≤ 15). The integral of t^2 - t - 182 is (1/3)t^3 - (1/2)t^2 - 182t. Evaluating this integral from 1 to 15 gives the displacement as follows:
Displacement = [(1/3)(15)^3 - (1/2)(15)^2 - 182(15)] - [(1/3)(1)^3 - (1/2)(1)^2 - 182(1)] = 2025 - 1125 - 2730 + 91 - (-182) = -2301 feet
(b) Total distance is the sum of the absolute values of the displacement. To find the total distance, we take the absolute value of the displacement:
Total distance = | -2301 | = 2301 feet