A reservoir begins releasing water at noon

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

Problem

A reservoir begins releasing water at noon. The height

H

(in meters) of the water above a sensor is modeled for the next eight hours by the piecewise function

H(t)=\begin{cases} -0.04t^2+1.6t+12, & 0\le t\le 4,\\ 14e^{-0.3(t-4)}, & 4<t\le 8,\end{cases}

where

t

is the number of hours after noon. (1) Verify whether

H

is continuous at

t=4

. (2) Assuming

H

is continuous on

[0,8]

, justify that there is at least one time

\tau

(0,8)

at which the water height was exactly

13

meters. (3) Explain why the conclusion in part (b) would fail if the release protocol created a jump discontinuity at

t=4