A settling pond has vertical cross–section bounded below by the concrete floor and…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A settling pond has vertical cross–section bounded below by the concrete floor y=0 and above by the water surface y=5 (units in meters). The right bank of the pond is modeled for 0≤y≤5 by the curve x=10−0.2y2. The left bank is described piecewise:
• x=0 for 0≤y≤1 (a vertical retaining wall)
• x=0.5(y−1) for 1≤y≤5
Between y=2 and y=3 there is a rectangular concrete pillar that occupies the horizontal interval 4≤x≤6. This pillar displaces water and must be excluded when computing the cross–sectional area available for water storage.
(a) Write a complete set of definite integrals that gives the area of the pond’s cross–section, excluding the portion occupied by the pillar. Use separate integrals for each portion where the boundary description changes or where the pillar is removed.
(b) Evaluate the integrals from part (a) to determine the exact cross–sectional area, in square meters, of the water–holding region.
(c) Explain, with reference to the geometry of the situation, why subtracting the pillar’s area from the overall region yields the desired result.