A river’s cross-sectional area at a particular gauging station can be modeled as a…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A river’s cross-sectional area at a particular gauging station can be modeled as a function of the water depth d (in meters) by
A(d)=15d2.5
where A(d) is measured in square meters.
Assume that, at a certain instant, the depth is exactly 3.00 m and is increasing over time.
(a) Derive an explicit expression for A′(d). Include appropriate units in your answer.
(b) Evaluate A′(3.00) and interpret its numerical value in the context of the river flow.
(c) At the instant when d=3.00 m, the depth is observed to be rising at a rate of 0.020 m min−1. Determine the instantaneous rate of change of the cross-sectional area with respect to time at that moment. State units clearly.
(d) Using a linear approximation based on your work in part (a), estimate the additional cross-sectional area created when the depth increases from 3.00 m to 3.05 m. Comment on the reliability of this approximation for the stated interval.