Determining Components of an Exponential Function \( \begin{array}{l}\text { A doctor estimates that a particular patient is losing bone density at a rate of } 3 \% \text { annually. The patient currently has a } \\ \text { bone density of } 1,500 \mathrm{~kg}_{\mathrm{m}} \mathrm{mg}^{3} \text {. The doctor writes an exponential function to represent the situation. Which values } \\ \text { should the doctor use for } a \text { and } b \text { in a function written in the form } f(x)=a b^{x} \text {, where } f(x) \text { represents the bone density } \\ a=\square \\ b=\square\end{array} \)
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The doctor can choose the current bone density, 1,500 kg/m³, as the initial value \( a \). This represents the baseline from which the bone density will decline each year. Hence, \( a = 1500 \). For the growth factor \( b \), since the bone density decreases by 3% annually, the remaining percentage after one year is 97%, or 0.97 in decimal form. Thus, \( b = 0.97 \), reflecting the reduced density each year. In function form, this gives us \( f(x) = 1500 \cdot (0.97)^x \). This function allows for a fun exploration of exponential decay and the impact of persistence over time!
