The function is r(t)= 400texp(-0.2t^2) and it shows the rate at which people show up in a line outside a theatre to buy tickets. t is the number of hours after 8:00am
Assume there is no people at 8:00am and the patrons are served at a constant rate of 200 people after the ticket window open at 9:am.
- Find the time at which the number of people of people in line is greatest.
-How many people eventually show up to buy the ticket.
-Find the time at which the queen completely disappears. (This is tricky, but interesting....)
Response to last quest, there is a typo. It should be the line instead of the queen
To find the time at which the number of people in line is greatest, we need to determine the maximum value of the function r(t).
1. Find the derivative of the function r(t) with respect to t.
The derivative of r(t) is given by:
r'(t) = d/dt (400t * e^(-0.2t^2))
= 400 * (e^(-0.2t^2) - 0.2t^2 * e^(-0.2t^2))
2. Set the derivative equal to zero to find the critical points:
r'(t) = 0
400 * (e^(-0.2t^2) - 0.2t^2 * e^(-0.2t^2)) = 0
Now, solve this equation to find the values of t at which the derivative is equal to zero. These values will correspond to local maxima or minima.
3. Determine the nature of the critical points:
To determine whether these critical points are local maxima or minima, we need to analyze the second derivative of r(t).
The second derivative of r(t) is given by:
r''(t) = d^2/dt^2 (400t * e^(-0.2t^2))
= 400 * (e^(-0.2t^2) - 0.2t^2 * e^(-0.2t^2)) * (-0.2t^2 * -0.2 + (-0.2) * e^(-0.2t^2)) + 400 * (2 * 0.2t * e^(-0.2t^2)) = 0
Simplify this equation to find the critical points.
4. Analyze the second derivative:
For each value of t obtained in step 3, evaluate the second derivative r''(t). If it is positive, the corresponding t value represents a local minimum, and if it is negative, it represents a local maximum.
5. Determine the time at which the number of people in line is greatest:
Among the t values obtained in step 4, find the one that corresponds to a local maximum. This time value will represent the time at which the number of people in line is the greatest.
To find how many people eventually show up to buy tickets, we need to integrate the function r(t) over the desired interval.
6. Integrate the function r(t) from 0 (time of 8:00am) to the desired end time:
The integral of r(t) with respect to t, denoted as ∫[r(t) dt], will give us the total number of people who show up to buy tickets during the given time frame.
∫[r(t) dt] = ∫[400t * e^(-0.2t^2) dt] evaluated from 0 (time of 8:00am) to the desired end time.
Evaluate this integral to find the total number of people who show up to buy tickets.
To find the time at which the queue completely disappears, we need to find the value of t at which r(t) becomes zero.
7. Solve r(t) = 0 to find the time at which the queue completely disappears:
400t * e^(-0.2t^2) = 0
Solve this equation to find the time at which the queen (queue) completely disappears.
Note: Please provide the desired end time to perform steps 6 and 7 accurately.