A coastal inlet is dredged in such a way that the water depth (in meters) at a point…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A coastal inlet is dredged in such a way that the water depth

D

(in meters) at a point

P

along the centerline of the channel depends on the horizontal distance

x

(in kilometers) from the shoreline according to

D(x)=\begin{cases} 3x^{2}+1, & 0\le x\le1.5,\\ 7.5-2(x-1.5), & 1.5<x<3,\\ 1+\cos\Bigl(\tfrac{\pi}{2}(x-3)\Bigr), & 3\le x\le4. \end{cases}

Assume the dredged channel ends at

x=4\,\text{km}

. (1) Part 1: (a) Verify that

D

is continuous at both

x=1.5

and

x=3

. (2) Part 2: (b) Compute

D'(x)

on each interval where it exists, and determine all critical points of

D

[0,4]

. (3) Part 3: (c) For each critical point, decide whether it corresponds to a local maximum depth, a local minimum depth, or neither. Support your answer by referencing the sign of

D'(x)

on the relevant intervals. (4) Part 4: (d) Identify the global maximum depth attained on the entire channel segment

0\le x\le4

and the location(s) where it occurs. Justify your conclusion. (5) Part 5: (e) Interpret your findings from part (d) in practical terms for a cargo vessel planning to navigate the channel.

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