2. One of the processes in the preparation of pastries is the cooling of a special mixture of ingredients. The mixture is made at a temperature of $180^{\circ} \mathrm{C}$ and is immediately cooled in a room with a constant temperature of $20^{\circ} \mathrm{C}$. It has been found that the current temperature $T$ of the mixture after $t$ hours of cooling in a room with a constant temperature $c$ can be calculated using the formula $T=a \cdot b^{t}+c$.

a) Determine the constants $a$ and $b$, given that the mixture has a temperature of $180^{\circ} \mathrm{C}$ at the start $(t=0)$ and a temperature of $160^{\circ} \mathrm{C}$ after 1 hour of cooling.

b) How long after preparation does the mixture cool to $150^{\circ} \mathrm{C}$? Calculate to the nearest minute. Write down your answer.
2. At the beginning, the mixture has a temperature of $180^{\circ}$, so $180=a \cdot b^{0}+20$, from which we calculate $a=160$. After 1 hour of cooling, the mixture has a temperature of $160^{\circ}$, so $160=a \cdot b+20$, from which we get $b=\frac{7}{8}$, considering that $a=160$. Therefore, the formula is $T=160 \cdot\left(\frac{7}{8}\right)^{t}+20$.

From the equation $150=160 \cdot\left(\frac{7}{8}\right)^{t}+20$, it follows that $\frac{13}{16}=\left(\frac{7}{8}\right)^{t}$, and from this we get $t=\frac{\log \frac{13}{16}}{\log \frac{7}{8}}=1.555$, which is 1 hour and 33 minutes.

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-10.jpg?height=57&width=1636&top_left_y=2062&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-10.jpg?height=55&width=1636&top_left_y=2114&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-10.jpg?height=100&width=1642&top_left_y=2149&top_left_x=207)

The equation for $T=150, 150=160 \cdot\left(\frac{7}{8}\right)^{t}+20 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-10.jpg?height=131&width=1642&top_left_y=2296&top_left_x=207)

Answer: The mixture cools to $150^{\circ} \mathrm{C}$ 1 hour and 33 minutes after preparation. ................ 1 point