Assignment:
Calculus: Use integral calculus to solve differential equation problems.
Q1. The United States Census Bureau mid-year data for the population of the world in the year 2000 was 6.079 billion. Tkee years later, in 2003, it was 6.302 billion. (A billion, as used here, is one thous, million.)
Year 2000 2003
t 0 3
Population in billions, N 6.079 6.302
a) Write a differential equation that may be an appropriate model to represent the population growth of the world from 2000 to 2003.
b) Solve your differential equation, showing worlcing to justify your solution.
c) Evaluate any constants using the data given.
d) Use your model to estirnate the population of the world in the year 2020.
e) Usirig a different model, the United States Census Bureau predicts that the population of the world in 2020 will be 7.515 billion. What is the difference between your prediction and the Census Bureau's prediction?
Q2. A metal ball, initially at a temperature of 90° C, is Unmersed in a large body of water at a temperature of 30°C. According to Newton's law of cooling, the temperature, T, of the ball t minutes after it is imtnersed in the water satisfies the
differential equation. dT/dt= -k(T- 30) where k is a constant.
a) Show that T =30+60e-kt. is a solution of the above differential equation, and this solution gives the correct temperature at t = O.
b) Five minutes later the temperature of the ball is 46°C. What will the temperature of the ball be after 10 minutes?
Provide complete and step by step solution for the question and show calculations and use formulas.