An irrigation channel has a cross‐section in the shape of a composite region bounded…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
An irrigation channel has a cross‐section in the shape of a composite region bounded above by the graph of y=5x+1, below by y=21x+1, and on the left and right by the vertical lines x=0 and x=4, respectively. However, an inspection hatch occupying the rectangular region 0≤x≤1 and 1≤y≤2 is cut away and does not hold water. Engineers wish to know the exact area of the water‐holding portion of the cross‐section.
(a) Determine every point of intersection among the curves y=5x+1, y=21x+1, and the lines x=0, x=4, y=1, and y=2 that are relevant to the description.
(b) Write, but do not yet evaluate, the integral expression(s) that give the exact cross‐sectional area available for water storage, accounting for the removal of the inspection hatch.
(c) Evaluate the integral(s) from part (b) to compute the exact area, expressed in square meters.
(d) If regulations require at least 9 m2 of cross‐sectional area, determine whether the channel meets the requirement and explain your reasoning.