Question

a doctor estimates that a particular patient is losing bone density at a rate of 3% annually. the patient currently has a bone density of 1,500 kg/mg³. the doctor writes an exponential function to represent the situation. which values should the doctor use for a and b in a function written in the form f(x) = abˣ, where f(x) represents the bone density after x years?
a =
b =

Explanation:

Step1: Determine the value of \(a\)

In the exponential function \(f(x)=ab^{x}\), \(a\) represents the initial value. The patient currently has a bone density of \(1500\space kg/mg^{3}\), so the initial value \(a = 1500\).

Step2: Determine the value of \(b\)

The patient is losing bone density at a rate of \(3\%\) annually. This means the remaining bone density each year is \(100\% - 3\%=97\% = 0.97\). In the exponential function, \(b\) is the growth (or decay) factor. Since it's a decay situation, \(b = 0.97\).

Answer:

\(a=\boldsymbol{1500}\)
\(b=\boldsymbol{0.97}\)