For , the depth (in meters) of water at a coastal dock is modeled by the piecewise…

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

For

0\le t\le 24

, the depth (in meters) of water at a coastal dock is modeled by the piecewise function

D(t)=\begin{cases} A t+10, & 0\le t\le 6, \\ B(t-6)^2+C(t-6)+D, & 6<t\le 18. \end{cases}

Time

t

is measured in hours after midnight. Measurements show that the water depth is

10

meters at midnight (

t=0

) and again at

6\,\text{PM}

(

t=18

). In addition, the rate of change of the depth at

6\,\text{PM}

-1\,\text{m/hr}

, indicating that the tide is receding at that moment. (a) Use the given observations to write three independent equations involving the constants

A

B

C

, and

D

. Clearly indicate which physical requirement (value or rate) generates each equation. (b) Solve the equations from part (a) to determine

A

B

C

, and

D

. Show all necessary algebraic steps. (c) Verify explicitly that the resulting model imposes no sudden jump in either the water level or its instantaneous rate of change at

t=6\,\text{h}

. State the common value of

D'(6)

from both sides.

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