Calc BC Name
WS 6.3 Logistic Equations Date Hour
1. a. The population of the world was about 5 billion in 1986. Using an exponential growth model with an observed rate of population increase of 2% per year in 1986, find an expression for the population of the world in year t.
b. Predict the population of the world in year 2000, 2100, and 2500.
c. The total land surface area of this planet is about .
Based on the predictions in part b, how much land will there be on average per person in each of those years?
2. Based on the numbers from problem 1, make the same predictions using a logistical model for each of those years if the carrying capacity of the world is M=100 billion.
3. One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction of y of the population who have heard the rumor and the fraction who have not heard the rumor.
a. Write a differential equation that is satisfied by y.
b. Solve this differential equation.
c. A small town has 1000 inhabitants. At 8 AM 80 people have heard the rumor. At noon, half of the town has heard it. At what time will 90% of the town have heard the rumor?
4. Biologists stocked a lake with 400 fish and estimate the carrying capacity to be 10000. The number of fish tripled in the first year.
a. Write a logistic equation equation to model the fish population at time t.
b. How long will it take for the fish population to reach 5000?
5. The Museum of Science in Boston displays a running total of the US population. On May 11, 1993, the total was increasing at a rate of 1 person every 14 seconds. The display population figure at 3:45 PM that day was 257,313,431. Assume M = 540.7 million people.
a. Find the relative growth rate per year (365 days)
b. What will the US population be at 3:45 PM EDT on May 11, 2001?
6. A 2000 gallon fish tank can support no more than 150 guppies. 6 guppies are introduced into the tank. Assume that the growth rate is where t is time in weeks.
a. Find the logistic model for the differential equation.
b. How long will it take for the guppy population to reach 100? 125?
7. A certain wild animal preserve can support no more than 250 lowland gorrilas. 28 gorillas were known to be in the preserve in 1970. Assume the growth rate is where t is time in years.
a. Find the logistic model for the differential equation.
b. How long will it take for the gorilla population to reach the carrying capacity of the preserve?