The temperature of water in a tea cup, $T\ ^\circ\text{C}$, $t$ minutes after boiling water is poured into the empty cup can be modelled by the formula $T = 25 + 75\text{e}^{-kt}$. The diagram above shows the ideal brewing temperatures for various teas at a particular teahouse. At this teahouse, green tea leaves are placed into the tea cup for brewing at the ideal temperature when $t = 6$. Determine whether the teahouse's standards for brewing tea would be adhered to if black tea leaves are placed into the teacup when $t = 2$.

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.