Question

Newton's law of cooling indicates that the temperature of a warn object will decrease exponentially with time and will approach the temperature of the

Exponential models
Newton's law of cooling indicates that the temperature of a warn object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T(t) is modeled by $$\displaystyle{T}{\left({t}\right)}={T}_{{a}}+{\left({T}_{{0}}-{T}_{{a}}\right)}{e}^{{-{k}{t}}}$$. In this model, $$\displaystyle{T}_{{a}}$$ represents the temperature of the surrounding air, $$\displaystyle{T}_{{0}}$$ 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 the object.
A cake comes out of the oven at 335F and is placed on a cooling rack in a 70F 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) of the cake t minutes after being removed from the oven.

2021-02-10

Given:
The initial temperature (T0 ) of the cake is $$335^{\circ}F$$.
The tempearture (Ta) of the cake when place in cooling tray is $$70^{\circ}F$$.
The cooling rate (k) is 0.050.
Known fact:
By the Newton's law of cooling , the temperature is modeled as
$$\displaystyle{T}{\left({t}\right)}={T}_{{a}}+{\left({T}_{{0}}-{T}_{{a}}\right)}{e}^{{-{k}{t}}}$$
where, $$\displaystyle{T}_{{a}}$$ is the temperature surronded by air.
$$\displaystyle{T}_{{0}}$$ is the initial temperature
t is the time after the object starts cooling,
k is the cooling rate.
Calculation:
The function that models the temperature of the cake t minutes after being removed from the oven is computed as follows.
Substitute
$$\displaystyle{T}_{{0}}={335}{F},{T}_{{a}}={70}{F}$$ and $$k=0.050 \in$$ $$\displaystyle{T}{\left({t}\right)}={T}_{{a}}+{\left({T}_{{0}}+{T}_{{a}}\right)}{e}^{{-{k}{t}}}$$
$$\displaystyle{T}{\left({t}\right)}={70}+{\left({335}-{70}\right)}{e}^{{-{\left({0.050}\right)}{t}}}$$
$$\displaystyle={70}+{265}{e}^{{-{\left({0.050}\right)}{t}}}$$
The function that models the temperature of the cake t minutes after it is removed from the oven is $$\displaystyle{T}{\left({t}\right)}={70}+{265}{e}^{{-{\left({0.050}\right)}{t}}}$$