Colonization
The initial group of colonists from country X sets sail on a boat to reach a newly discovered island. The boat can travel at a constant speed of 20 km/h initially and accelerates at a rate of .5 km/h^2. If it takes 3 days to reach the island, how far is the island from the starting point?
56 km/hr
Colony X discovers a valuable resource and begins exporting it to other regions. The rate of export is directly proportional to the current stockpile of the resource. If the initial stockpile is 1000 units and dy/dt = 50y, write a differential equation to model the change in the stockpile of the resource over time.
y = 1000e^50t
If the closed curve in the xy-plane given by x^2 + 3x + y^4 + 5y = 6 represents a circular missile launched by an alien army. Write an equation for when the land(represented as a line) is tangent to the curve at point (2,1).
y = -7/9 (x-2) +1
During the journey, some of the passengers on the boat fall sick. The rate at which sick passengers are thrown out of the boat is proportional to the number of healthy passengers on board. If there are initially 200 healthy passengers on the boat, and the rate constant is 0.02 per day, write a differential equation to model the number of healthy passengers on the boat as a function of time.
y = 200e^(0.2t)
After Colony X discovered gold, the colony is experiencing exponential population growth at a rate of 15% per year. If the initial population is 1000, find the population after 10 years. Round to the nearest person.
4046 people
The radius, r km, of a circular bomb, t seconds after it was detonated, satisfies the differential equation dr/dt = k/r , r ≠ 0, where k is a positive constant. The initial radius of a bomb explosion is 8 km and 16 seconds later, it has increased to 40 km. Find the time when the stain will have a circumference of 112π.
44.71 s
After arriving on the island, the colonists begin establishing a colony named X. The population of the colony is initially 500 individuals and is growing at a rate proportional to its current population. The growth rate constant is estimated to be 0.003 per day. Find the population at one year. Round to the nearest person.
1495 people
Colony X is expanding its territory to accommodate the growing population. The rate of land expansion is proportional to the square root of the current population. Write a differential equation to model the increase in land area over time, given that the initial land area is 500 square kilometers and the land area increases to 675 in 1 year.
y = 1000e^50t
The blast has sent radioactive particles throughout the land. 200 grams of a radioactive substance was sent out. But only 64 grams remained 20 years later. Find the substance’s half-life using a differential equation.
12.1665 s
Unfortunately, the colony faces challenges, and the death rate in the colony increases from .0005 per month to .0006 per month in 3 months. Write a differential equation to model the death of the colony population over time. Find the rate constant and how long does it take for the death rate to reach .001 per day.
11.4 months
Technological advancements in Colony X are creating job opportunities at a rate proportional to the derivative of the population with respect to time. Write a differential equation to model the change in the number of job opportunities over time, given that the population is growing exponentially at a rate of 3% per year and there are initially 500 job opportunities. How many more job opportunities are available after 5 years
80 more jobs
The blast has sent harmful chemicals in the air that citizens are now breathing and dying from. A sample shows that 50 thousand people remain in the area and after 2.5 hours 35 thousand people remain. Assuming the death rate is proportional to the population, when does the population reach 5 thousand?
16.13924 hr
The island experiences a natural disaster, which affects the growth of the colony. The growth rate constant decreases from 2 people/month to 1.8 people/month after 5 months. Determine the number of months it takes for population growth to reach .5 people/month.
65 months
A small island nation is holding a crucial vote on immigration policy. The rate at which people from the island travel to the mainland to vote is directly proportional to the difference between the current population on the island and the mainland. Additionally, the rate of travel is inversely proportional to the travel time required. Given the equation dy/dt = .3y, which represents the rate of travel to the island, what is the differential equation if at t=5 there are 5036 people traveling each year
y=1123.683e^.3t
The remaining citizens have employed robots like Wall-E to clean up the radioactive waste remaining. The robots in Area A have to clean up 12,000 kg of radioactive waste in that area. On the first day, they cleaned up 6 kg of waste. 6 days later, they cleaned up 600 kg of waste. Suppose that the robots clean at a proportional rate to both the number of waste cleaned and the number of waste not cleaned. Find the number of days for 50% of the waste to be cleaned up.
81.0804 days