The cooling process in the first 10 minutes is to be modeled for $0 \leq t \leq 10$ by a function $u _ { 1 }$ defined on $\mathbb { R }$ with $u _ { 1 } ( t ) = a + b \cdot \mathrm { e } ^ { - c \cdot t } , a , b , c \in \mathbb { R } , c > 0$. Here, $t$ denotes the time in minutes since the beginning of the investigation and $u _ { 1 } ( t )$ denotes the temperature of the coffee in ${ } ^ { \circ } \mathrm { C }$. Explain why $a = 18$ is chosen in the modeling, and then calculate the values of $b$ and $c$ based on the information in the table. [Check solution with rounded values: $u _ { 1 } ( t ) = 18 + 55,86 \cdot \mathrm { e } ^ { - 0,053 \cdot t }$.]
The cooling process in the first 10 minutes is to be modeled for $0 \leq t \leq 10$ by a function $u _ { 1 }$ defined on $\mathbb { R }$ with $u _ { 1 } ( t ) = a + b \cdot \mathrm { e } ^ { - c \cdot t } , a , b , c \in \mathbb { R } , c > 0$.
Here, $t$ denotes the time in minutes since the beginning of the investigation and $u _ { 1 } ( t )$ denotes the temperature of the coffee in ${ } ^ { \circ } \mathrm { C }$.
Explain why $a = 18$ is chosen in the modeling, and then calculate the values of $b$ and $c$ based on the information in the table.
[Check solution with rounded values: $u _ { 1 } ( t ) = 18 + 55,86 \cdot \mathrm { e } ^ { - 0,053 \cdot t }$.]