The people move toward the door at speed vs = 3.7 m/s, are each d = 0.20 m in depth, and are separated by L = 1.75 m. The arrangement in the figure occuurs at time t = 0.

(a) At what average rate the layer of people at the door increase?
m/s

(b) At what time does the layer's depth reach 5.9 m? (The answer reveals how quickly such a situation becomes dangerous.)
s

(a) To find the average rate at which the layer of people at the door increase, we need to calculate the change in depth over time.

The rate at which the layer of people at the door increase can be calculated as follows:

Rate = Change in depth / Time

The change in depth can be determined by the formula:

Change in depth = Depth of each person x Number of people

In this case, the depth of each person is given as d = 0.20 m, and the people are separated by L = 1.75 m.

To find the number of people, we can divide the total length covered by the people by the distance between them:

Number of people = Length covered / Distance between people

Given that the speed of people moving toward the door is vs = 3.7 m/s, and they start moving at time t = 0, we can determine the length covered by the people at any given time as:

Length covered = vs x t

Substituting the given values, we can now calculate the number of people:

Number of people = (vs x t) / L

Finally, we can calculate the change in depth:

Change in depth = d x (Number of people)

Now, we can calculate the average rate at which the layer of people at the door increase:

Rate = Change in depth / Time

(b) To find the time at which the layer's depth reaches 5.9 m, we need to set up the equation and solve for time.

Using the formula we derived above:

Length covered = vs x t

Setting this equal to 5.9 m:

vs x t = 5.9

Solving for time:

t = 5.9 / vs

Substituting the given value of vs = 3.7 m/s:

t = 5.9 / 3.7

Now, we can calculate the time at which the layer's depth reaches 5.9 m.