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

question 120

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 \(106 ^ { \circ } \mathrm { F }\)

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Definitions:

Population Variance

A measure of the dispersion of a set of data points in a population, showing how much the data deviates from the mean of the population.

Population Proportion

A measure indicating the ratio of members in a defined category to the total population size.

Interval Estimate

A range of values derived from sample data within which a population parameter is estimated to lie, usually defined by two numbers representing the upper and lower limits.

Population Standard Deviation

A measure of the variability or dispersion of a population dataset, representing the square root of the variance.