3. A doctor sees between 7 and 12 patients each day. On Mondays and Tuesdays, the appointment times are 15 minutes. On Wednesdays and Thursdays, they are 30 minutes. On Fridays, they are one hour long. The doctor works for no more than 8 hours a day. Here are some inequalities that represent this situation. \[ 0.25 \leq y \leq 1 \] a. What does each variable represent?

simplify: \( \frac{3-3 \sqrt{3 a}}{4 \sqrt{8 a}} \)

a) \( \frac{3}{4}+\frac{1}{6}= \)

8.) Line \( l \) passing through the points \( (-3,-5) \) and \( (0,-5) \) Line \( n \) passing through the points \( (0,5) \) and \( (4,9) \)

Avalanche forecasters measure the temperature gradient \( \frac{d T}{d h} \), which is the rate at which the temperature in a snowpack T changes with respect to its depth h . A large temperature gradient may lead to a weak layer in the snowpack. When these weak layers collapse, avalanches occur. Avalanche forecasters use the rule of thumb that if \( \frac{d T}{d h} \) exceeds \( 10^{\circ} \mathrm{C} / \mathrm{m} \) anywhere in the snowpack, conditions are favorable for weak-layer formation, and the risk of avalanche increases. Assume the temperature function is continuous and differentiable. Complete parts (a) through (d) below. a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h=0), the temperature is \( -10^{\circ} \mathrm{C} \). At a depth of 1.8 m , the temperature is \( -1^{\circ} \mathrm{C} \). Using the Mean Value Theorem, what can he conclude about the temperature gradient? The average temperature gradient from \( \mathrm{h}=0 \) to \( \mathrm{h}=1.8 \) is \( \square{ }^{\circ} \mathrm{C} / \mathrm{m} \). (Simplify your answer. Round to the nearest tenth as needed.)

Determine whether Rolle's theorem applies to the function shown below on the given interval., If so, find the point(s) that are guaranteed to exist by Rolle's theorem. \( f(x)=x(x-8)^{2} ;[0,8] \)