6.3: Logarithms and exponential models

1. The population of a region (in thousands of people) is \(P=3.4 \times 1.023^t t\) years after a census taken in 2000.

1. What is the annual percent growth rate of the population?
   
   
   
   
    
2. How long does it take for the population to double? This is called, naturally, the doubling time.
   
   
   
   
    
3. Is the doubling time constant? That is, does the population take the same time to double from any starting point? Discuss with your group, and write an explanation of your answer in a sentence or two.

2. The radioactive isotope cobalt-60 has a half-life of 5.27 years.

1. Use the half-life to write an expression for \(f(t)\), the amount of cobalt-60 left after \(t\) years. (Hint: set up an equation \(\frac{9_0}{2}=Q_0 \times b^{5.27}\) and solve.)
   
   
   
   
    
2. If a sample contains 20 g of cobalt-60 today and is placed in storage for 30 years, how much cobalt-60 should be in the sample at the end of that time? Write an expression and evaluate with a calculator.
   
   
   
   
    
3. A sample containing cobalt-60 has been in storage for 10 years, and is found to contain 12g of the isotope. How much did it have before being placed in storage? You will need to set this up as an equation.
   
   
   
   
    
4. How long does it take for \(90 \%\) of the isotope in a sample to decay?
   
   
   
    

3. Consider the exponential function \(g(t)=12 \times 1.247^{\mathrm{t}}\).

1. Rewrite the expression for \(g(t)\) in the form \(g(t)=a e^{k t}\). (Hint: you can figure out the value of \(a\) without an equation, then solve an equation to find \(k\).)
   
   
   
   
    
2. Give the growth rate and the continuous growth rate for \(g\).
   
   
   
   
    

4. Suppose the quantity of a pollutant in a lake is decreasing exponentially at a continuous rate of 3.4\% per day. Find the non-continuous rate of decay, giving your answer as a percentage.

 

 

 

5. According to Newton's law of cooling, the temperature of an object that has been cooling for an amount of time \(t\) is given by the formula

\[
T(t)=T_a+\left(T_0-T_a\right) e^{-k t}
\]

where \(T_0\) is the initial temperature, \(T_{\mathrm{a}}\) is the ambient temperature of its surroundings, and \(k\) is a constant that depends on the object and where it is.

The same formula works for warming.

A pie is taken out of the oven, and its temperature is \(210^{\circ} F\). It is set out on the counter in a room where the air is 70 degrees F.
 

1. After 10 minutes the pie has cooled to 200 . Use this to find the constant \(k\) for this situation.
   
   
   
   
    
2. How long will it take for the pie to cool to \(150^{\circ} F\) ?

 

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