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Sagot :
So the proportion of women who completed at least 4 years of college since 1940 is given by:
[tex]D(t)=0.036\cdot(1.031)^t[/tex]Where t represents the years past since 1940. We need to find the t value from which this proportion is equal or greater than 0.04. What we basically need to do is take D(t)=0.04 and then solve the equation for t. Then we get:
[tex]\begin{gathered} 0.04=0.036\cdot(1.031)^t \\ \frac{0.04}{0.036}=1.031^t \\ \frac{10}{9}=1.031^t \end{gathered}[/tex]In order to continue we can use a property of logarithmic functions. Remember that:
[tex]\log (a^b)=b\cdot\log (a)[/tex]So let's apply a logarithm at both sides of the equation:
[tex]\begin{gathered} \frac{10}{9}=1.031^t \\ \log (\frac{10}{9})=\log (1.031^t) \\ \log (\frac{10}{9})=t\cdot\log (1.031) \\ t=\frac{\log (\frac{10}{9})}{\log (1.031)}=3.45 \end{gathered}[/tex]This means that 3.45 years after 1940 the proportion of the US population of women who have completed at least four years of college is 0.04. And that's the complete answer.
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