To represent the situation where a patient is losing bone density at a rate of [tex]$3 \%$[/tex] annually, with the current bone density being [tex]$1,500 \, \text{kg/mg}^3$[/tex], we need to form an exponential function of the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( f(x) \)[/tex] represents the bone density after [tex]\( x \)[/tex] years.
### Step-by-Step Solution:
1. Identify the initial bone density:
- The patient currently has a bone density of [tex]\( 1,500 \, \text{kg/mg}^3 \)[/tex].
- This value represents [tex]\( a \)[/tex], the initial value of the bone density.
2. Determine the annual decay factor:
- The patient is losing bone density at a rate of [tex]\( 3 \% \)[/tex] annually.
- A [tex]\( 3 \% \)[/tex] loss means that [tex]\( 97 \% \)[/tex] of the bone density is retained each year.
- Therefore, the decay factor [tex]\( b \)[/tex] will be [tex]\( 1 - 0.03 = 0.97 \)[/tex].
3. Assign values to [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\( a = 1500 \)[/tex]
- [tex]\( b = 0.97 \)[/tex]
By using these identified values:
[tex]$
\begin{array}{l}
a = 1500 \\
b = 0.97
\end{array}
$[/tex]
Therefore, the exponential function [tex]\( f(x) = a \cdot b^x \)[/tex], which represents the bone density after [tex]\( x \)[/tex] years, should be formulated with [tex]\( a \)[/tex] and [tex]\( b \)[/tex] as follows:
- [tex]\( a = 1500 \)[/tex]
- [tex]\( b = 0.97 \)[/tex]
