Fill Needed for a Highway Entrance Ramp

by Caitlin McHale

Submitted : Spring 2013

Given the instructions to determine the volume of fill and concrete area needed for a highway entrance ramp, one must complete a series of integrations. For part 1, after determining which coordinate axis to use, the volume is calculated using a triple integral with dV equal to r*dz*dr*dθ. Then, the boundaries for the integral are found using information from the problem as well as further calculations. Once the boundaries are determined, the integral is solved to completion, with an ending result having the units meters³.

Next, for part 2, the surface area is found using, instead, a double integral. Two integrals are necessary being that there is an outside area, having radius (R+W), and an inside area having radius R. Both integrals are performed over the integration of r*dθ*dz, substituting the corresponding radius value for r. The boundaries for this integration are: θ equals the given values from the problem, and z equals the height of the ramp at the two radii (z₀=0 and z_f=h(θ) for the inside and z_f= h(θ)+wtanφ for the outside). The two integrals are then calculated to completion, giving a final answer having the units meters².

After obtaining the ending results containing the parameters R, W, H, and φ, the values from part 3 are substituted to give a final answer for the volume and surface area of the ramp.

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