Solve the Problem $T ( t )$ Is Modeled By $T ( t ) = T _ { a } + \left( T _ { 0 } - T _ { a } \right) e ^ { - k t }$

Multiple Choice

Solve the problem.
-Newton's law of cooling indicates that the temperature of a warm object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature $T ( t )$ is modeled by $T ( t ) = T _ { a } + \left( T _ { 0 } - T _ { a } \right) e ^ { - k t }$ . In this model, $T _ { a }$ represents the temperature of the surrounding air, $T _ { 0 }$ represents the initial temperature of the object and $t$ is the time after the object starts cooling. The value of $k$ is the cooling rate and is a constant related to the physical properties of the object.
Water in a water heater is originally $131 ^ { \circ } \mathrm { F }$ . The water heater is shut off and the water cools to the temperature of the surrounding air, which is $80 ^ { \circ } \mathrm { F }$ . The water cools slowly because of the insulation inside the heater, and the rate of cooling is $0.00348$ . Dominic does not like to shower with water less than $116 ^ { \circ } \mathrm { F }$ . If Dominic waits $24 \mathrm { hr }$ after the heater is shut off, will the water still be warm enough for a shower? How warm will the water be?

A) No; the temperature after $24 \mathrm { hr }$ will be approximately $80 ^ { \circ } \mathrm { F }$
B) Yes; the temperature after $24 \mathrm { hr }$ will be approximately $128 ^ { \circ } \mathrm { F }$
C) Yes; the temperature after $24 \mathrm { hr }$ will be approximately $127 ^ { \circ } \mathrm { F }$
D) $\mathrm { No }$ ; the temperature after $24 \mathrm { hr }$ will be approximately $106 ^ { \circ } \mathrm { F }$