1. A population of rats follow the logistic model of population growth. The carrying capacity of the population is 10,000. The initial population is 1,500 and after 5 weeks, it is 4,000.
a. Write the logistic differential equation for the given data.
b. Find the formula for the weekly growth of population. What is the population in 10 weeks? When will the population be 9,000?
2. The rate at which an epidemic spreads through a community with 2000 susceptible residents is jointly proportional to the number of residents who have been infected and the number of susceptible residents who have not yet been infected by the disease. Suppose that 500 residents had the disease initially and 855 residents had been infected by the end of the first week. After how many weeks will 1000 residents be infected?
3. A freshly brewed cup of coffee has temperature 95 degrees Celsius in a 20 degrees Celsius room. When its temperature is 70 degrees Celsius, it is cooling at the rate of 1 degree Celsius per minute. What is the temperature of the coffee after t seconds?
4. A population grows weekly according to the logistic law. At the start, there are 200 individuals in the population. Write the differential equation describing the rate of change of population per week if the carrying capacity of the population is 3000 individuals and k=0.06. Find the population after 30 weeks.
5. One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5,000 inhabitants. 160 people have a disease at the beginning of the week and 1,200 have it at the end of the week. How long does it take for 80% of the population to become infected?
Sunday, March 23, 2014
Differential Equations Word Problems
1. In a town of population 12000, the rate of growth of a flu epidemic is jointly proportional to the number of people who have the flu and the number of people who do not have it?
a. If five days ago, 400 people in the town had the flu and today, 1000 people have it, find a mathematical model describing the epidemic.
b. How many people are expected to have the flu tomorrow?
c. In how many days will half the population have the flu?
d. Show that if the epidemic is not halted, the entire population will have the flu within three and one half months.
2. A cup of coffee is served at 180 degrees Fahrenheit in a room of temperature 60 degrees Fahrenheit. After one minute, the temperature dropped to 176 degrees Fahrenheit. Find the temperature at any time and determine when the temperature will reach 120 degrees Fahrenheit.
3. Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program, which included plans to make the college coeducational. The spread of the news follow a logistic equation and the total population of the college is 3000. Let N be the number of students who learned of the new program t hours later.
a. Write a differential equation that models the spread of the news.
b. If 600 students on campus had heard about the news 2 hours after the ceremony, how many students had heard about it after 4 hours?
4. A tank contains 100 liters of brine, and the brine has 4 kg of dissolved salt. Suppose that brine containing 2 kg of salt per liter is allowed to enter the tank at the rate of 5 liters/min and that the mixed solution is drained from the tank at the same rate. Find the amount of salt in the tank after 10 minutes.
5. A picture 5 ft high is placed on a wall with its base 7 ft above the eye level of an observer. If the observer is approaching the wall at the rate of 3 ft/sec, how fast is the measure of the angle subtended at her eye by the picture changing when she is 10 ft from the wall?
a. If five days ago, 400 people in the town had the flu and today, 1000 people have it, find a mathematical model describing the epidemic.
b. How many people are expected to have the flu tomorrow?
c. In how many days will half the population have the flu?
d. Show that if the epidemic is not halted, the entire population will have the flu within three and one half months.
2. A cup of coffee is served at 180 degrees Fahrenheit in a room of temperature 60 degrees Fahrenheit. After one minute, the temperature dropped to 176 degrees Fahrenheit. Find the temperature at any time and determine when the temperature will reach 120 degrees Fahrenheit.
3. Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program, which included plans to make the college coeducational. The spread of the news follow a logistic equation and the total population of the college is 3000. Let N be the number of students who learned of the new program t hours later.
a. Write a differential equation that models the spread of the news.
b. If 600 students on campus had heard about the news 2 hours after the ceremony, how many students had heard about it after 4 hours?
4. A tank contains 100 liters of brine, and the brine has 4 kg of dissolved salt. Suppose that brine containing 2 kg of salt per liter is allowed to enter the tank at the rate of 5 liters/min and that the mixed solution is drained from the tank at the same rate. Find the amount of salt in the tank after 10 minutes.
5. A picture 5 ft high is placed on a wall with its base 7 ft above the eye level of an observer. If the observer is approaching the wall at the rate of 3 ft/sec, how fast is the measure of the angle subtended at her eye by the picture changing when she is 10 ft from the wall?
Subscribe to:
Posts (Atom)