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

question 114

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.
A cake comes out of the oven at $335 ^ { \circ } \mathrm { F }$ and is placed on a cooling rack in a $70 ^ { \circ } \mathrm { F }$ kitchen. After checking the temperature several minutes later, it is determined that the cooling rate $k$ is $0.050$ . Write a function that models the temperature $T ( t )$ (in ${ } ^ { \circ } \mathrm { F }$ ) of the cake $t$ minutes after being removed from the oven.

• A) \(T ( t ) = 335 + 70 e ^ { 0.050 t }\)
• B) \(T ( t ) = 70 + 265 e ^ { 0.050 t }\)
• C) \(T ( t ) = 70 + 265 e ^ { - 0.050 t }\)
• D) \(T ( t ) = 70 + 335 e ^ { 0.050 t }\)

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Definitions:

Behavioural Inhibition

A temperament in early childhood that may be related to anxiety disorders later in life, characterized by profound avoidance of others in preschool, and atypical autonomic nervous system responses to novelty.

Anxiety Disorder

A mental health condition characterized by excessive fear or worry that interferes with daily activities.

Homotypic Continuity

An underlying (developmental) process or impairment may change, but the way it is manifested stays the same (e.g., the prediction of a disorder by the same disorder).

Incredible Years Parent Program

The Incredible Years Parent Program is a series of educational and supportive interventions aimed at improving parenting skills, strengthening parent-child interactions, and reducing children's behavior problems.