A tunnel’s ceiling is modeled by meters above ground level for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A tunnel’s ceiling is modeled by c(x)=10−0.05(x−20)2 meters above ground level for 0≤x≤40, where x is the horizontal distance (in meters) measured along the tunnel’s length. The road surface inside the tunnel rises according to r(x)=3+0.01x meters above the same reference level. Cross–sections of the tunnel perpendicular to its length are identical along the entire tunnel.
(a) Explain why there are exactly two points in the interval [0,40] where the road surface meets the ceiling. Support your reasoning with an analysis of the functions c and r.
(b) Approximate, to the nearest tenth of a meter, the x–coordinates where c(x)=r(x).
(c) Set up a definite integral whose value equals the cross–sectional area of the open space between the ceiling and the road surface. Clearly indicate the limits of integration.
(d) Evaluate the integral from part (c) to the nearest square meter. State your answer with appropriate units, and describe what this number represents in the context of the tunnel.