Extra Challenge Solutions

College Algebra

 

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

                                                                            

3.        The population of Canada is 30 million people and is growing at an annual rate of 1.4%.  The population of Germany is 80 million people and is decreasing at an annual rate of 1.7%.  In how many years will the population of Canada be equal to the population of Germany?  (Use logarithms to solve the resulting equation and answer accurate to 2 decimal places.)
         

For Canada y=30 e^(.014t) in millions
For Germany y = 80 e^(-.017t) in millions

When will populations be equal?
30 e^(.014t) = 80 e^(-.017t)

Divide both sides by 30 e^(.014t):
(30 e^(.014t))/(30e^(.014t)) = (80 e^(-.017t))/(30e^(.014t))
1 = (80 e^(-.017t))/( 30e^(.014t))

When you divide with the same base number, you subtract exponents:
1 = (8/3) e^(-.017t- .014t) = (8/3)*e^(-.031t)

Multiply both sides by 3/8
3/8 = e^(-.031t)
Take the ln of each side:
ln(3/8) = ln(e^(-.031t))
ln (3/8) = -.031t
t= (ln(3/8))/ -.031
t= 31.639655 calculator accuracy
Rounds off to 31.64 years.

 

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Dr. Robert J. Rapalje Altamonte Springs Campus
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