"Here a Lee, there a Lee... Social-Lee, Political-Lee, Financial-Lee"
The above is clearly a dramatization (a la 1776 the Musical) THIS, is what Richard Henry Lee most likely looked like:
Orange attire not included
Lee was instrumental in the adoption of the Declaration of Independence because he was from Virginia, one of the richest colonies at the time, which gave the opportunity to put forth the motion for the Declaration and gain support for it.
In order to gain Virginian support, he gave a speech to the House of Burgesses:
Before Lee arrived, there were already 700 people in the crowd. During the day, people joined the crowd according to this graph for a function r (t):
People also leave the crowd at a rate of 800 people per hour.
(a) How many people join the crowd between t = 0 and t = 3?
In order to understand how to approach this problem, you must realize that the graph they have given you relates the rate of people entering the crowd to time. The question asks for the amount of people that join the crowd, so therefore you must take the definite integral (or the area under the curve) from t = 0 to t = 3 to find out the amount of people that has joined the crowd from the rate graph they give you:
Since the integral is the area under the curve, using the area of a trapezoid formula would have net you the correct answer. Also note that the original 700 people that were already there do not contribute to the answer as the question asks for how many people ARRIVE.
(b) Is the number of people in the crowd increasing or decreasing between t = 2 and t = 3?
Due to Richard Henry Lee's charisma (and the graph), the size of the crowd is still increasing between the second and third hour. According to the graph, more than 800 people are joining the crowd between t = 2 and t = 3. Since this is greater than the 800 people per hour who are leaving, the size of the crowd must be getting larger.
(c) At what time "t" is the size of the crowd the largest? How many people are in the line at that time?
The point at which the crowd is the largest is when the size of the crowd goes from growing to shrinking. Since 800 people are leaving each hour, you must find the point on the graph at which the people joining the crowd is less than 800 which is at t = 3.
The second part of the question must be answered with a definite integral because it is asking you to find the amount people in the line from the graph of the rate at which people are joining the crowd. Now you must take into account the 700 people that were originally there and the 800 people that leave every hour. This means that you must subtract the definite integral from t = 0 to t = 3 of 800 from the definite integral of the graph and add 700:
(d) Write, but do not solve, an equation involving the integral expression of r whose solution gives the earliest time "t" for which there is no longer a crowd.
This follows the same principal as the previous problem. Add the integral of the graph, subtract the integral of the 800, and add 700. Remember to set your equation equal to 0 as that is what the question is asking:
His speech probably went something like this:
In order to gain Virginian support, he gave a speech to the House of Burgesses:
Before Lee arrived, there were already 700 people in the crowd. During the day, people joined the crowd according to this graph for a function r (t):
Eight hour speeches were the norm back in the day...
(a) How many people join the crowd between t = 0 and t = 3?
In order to understand how to approach this problem, you must realize that the graph they have given you relates the rate of people entering the crowd to time. The question asks for the amount of people that join the crowd, so therefore you must take the definite integral (or the area under the curve) from t = 0 to t = 3 to find out the amount of people that has joined the crowd from the rate graph they give you:
(b) Is the number of people in the crowd increasing or decreasing between t = 2 and t = 3?
Due to Richard Henry Lee's charisma (and the graph), the size of the crowd is still increasing between the second and third hour. According to the graph, more than 800 people are joining the crowd between t = 2 and t = 3. Since this is greater than the 800 people per hour who are leaving, the size of the crowd must be getting larger.
(c) At what time "t" is the size of the crowd the largest? How many people are in the line at that time?
The point at which the crowd is the largest is when the size of the crowd goes from growing to shrinking. Since 800 people are leaving each hour, you must find the point on the graph at which the people joining the crowd is less than 800 which is at t = 3.
The second part of the question must be answered with a definite integral because it is asking you to find the amount people in the line from the graph of the rate at which people are joining the crowd. Now you must take into account the 700 people that were originally there and the 800 people that leave every hour. This means that you must subtract the definite integral from t = 0 to t = 3 of 800 from the definite integral of the graph and add 700:
This follows the same principal as the previous problem. Add the integral of the graph, subtract the integral of the 800, and add 700. Remember to set your equation equal to 0 as that is what the question is asking:
His speech probably went something like this:
Questions? Concerns? Contact me at Jonathan352@gmail.com
