Question

the number of females in medicine continues to show steady increases. the number of female physicians in a certain country can be modeled by the function ( f(x)=0.673x^{1.211} ) where ( x ) is the number of years after 1970 and ( f(x) ) is the number of female physicians in thousands.
a. what type of function is this?
b. what is ( f(30) )? what does it mean?
c. how many female physicians will there be in 2040?
a. ( f(x)=0.673x^{1.211} ) is a power function.
b. ( f(30)=square ) (round to the nearest thousandth as needed.)

Explanation:

Step1: Substitute \(x = 30\) into the function

We have the function \(F(x)=0.673x^{1.211}\). When \(x = 30\), we substitute \(x\) into the function: \(F(30)=0.673\times30^{1.211}\).
First, calculate \(30^{1.211}\). Using a calculator, \(30^{1.211}\approx30^{\frac{1211}{1000}}\). Let \(y = 30^{1.211}\), take the natural logarithm of both sides: \(\ln y=1.211\ln30\). Since \(\ln30=\ln(3\times10)=\ln3+\ln10\approx1.0986 + 2.3026=3.4012\), then \(\ln y=1.211\times3.4012 = 4.1188\), and \(y = e^{4.1188}\approx61.57\).

Step2: Calculate \(F(30)\)

Now, \(F(30)=0.673\times61.57\). Using a calculator, \(0.673\times61.57=(0.6 + 0.07+0.003)\times61.57=0.6\times61.57+0.07\times61.57 + 0.003\times61.57=36.942+4.31 + 0.185=41.437\).

Answer:

\(F(30)\approx41.437\). It means that 30 years after 1970 (i.e., in the year 2000), the number of female physicians is approximately \(41.437\) thousand.