Finding Balance and Stability in a Skate Park Ramp

A local recreation department is building a new skate park, and you have been asked to help analyze one of the ramp components. The ramp’s side support panel is a triangular piece of reinforced material with vertices at [latex](0, 0)[/latex], [latex](4, 0)[/latex], and [latex](0, 3)[/latex], where distances are measured in feet. Due to the manufacturing process, the density of the material varies across the panel according to the function [latex]\rho(x,y) = 2 + x[/latex] lb/ft², with higher density near the outer edge providing extra durability where skaters apply the most force. Before installation, engineers need to determine the total mass of the panel, locate its center of mass for proper mounting, and calculate its moment of inertia to understand how it will respond to rotational forces during use. Your task is to perform these calculations to ensure the ramp is installed safely and securely.