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# Integral calculus to solve differential equation problems

1. The United States Census Bureau mid-year data for the population of the world in the year 2000 was 6.079 billion. Three years later, in 2003, it was 6.302 billion. Answer the following questions. (See attached bmp file)

2. A metal ball, initially at a temperature of 90 C, is immersed on 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 immersed in the water satisfies the differential equation dT/dt = -k(T-30) where k is a constant. Answer the following questions. (See attached bmp file)

See attached file for full problem description.

#### Solution Preview

1. a.

Population at a certain time t = N(t)
T in parenthesis means N is a function of t.

Population growth rate at any time t is proportional to the population at that time. In mathematical form one can write this as follows,

dN/dt &#945; N(t)

or introducing a constant r,

dN/dt = r N

b. Rearranging, dN/N = r dt , integrating both sides, ln N = r t + A where A is a constant. Let No is the population at time t = 0. Substituting this condition above,

ln No = 0 + A. Hence, A = ln No

so we get, ln N = r t + ln No or,
N/No = ...

#### Solution Summary

Very detailed solution is given for two lengthy word problems involving differential equations.

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