将沸水倒入空茶杯中 $t$ 分钟后，茶杯中水的温度 $T\ ^\circ\text{C}$ 可以用公式 $T = 25 + 75\text{e}^{-kt}$ 来建模。 上图显示了某茶馆中各种茶的理想冲泡温度。在这家茶馆，当 $t = 6$ 时，绿茶茶叶被放入茶杯中，在理想温度下进行冲泡。 确定如果当 $t = 2$ 时将红茶茶叶放入茶杯中，该茶馆的冲泡茶标准是否会得到遵守。

1. Set up the equation using the given information for green tea to find $k$.

From the diagram, the ideal temperature for green tea is $80^\circ\text{C}$.
When $t = 6$, $T = 80$.
$80 = 25 + 75\text{e}^{-6k}$

2. Solve for $k$.

$55 = 75\text{e}^{-6k}$
$\text{e}^{-6k} = \frac{55}{75} = \frac{11}{15}$
$-6k = \ln\left(\frac{11}{15}\right)$
$k = -\frac{1}{6}\ln\left(\frac{11}{15}\right) \approx 0.051686$

3. Calculate the temperature when $t = 2$.

Substitute $t = 2$ and $k \approx 0.051686$ into the formula:
$T = 25 + 75\text{e}^{-2(0.051686)}$
$T = 25 + 75\text{e}^{-0.10337}$
$T \approx 25 + 75(0.9018)$
$T \approx 92.63^\circ\text{C}$

4. Compare the calculated temperature with the ideal range for black tea and conclude.

From the diagram, the ideal temperature range for black tea is $90 - 95^\circ\text{C}$.
Since $92.63^\circ\text{C}$ is between $90^\circ\text{C}$ and $95^\circ\text{C}$, the teahouse's standards would be adhered to.