Question

personnel expenses for a small city were $9,200,000 in 2014. over the previous 5 years, the personnel expenses increased at a rate of 3.6% annually. an equation describing personnel costs in millions of dollars is c(t)=9.2(1.036)^t. what is the c - intercept? what is the practical meaning of the intercept in this situation?
the c - intercept is
(type an ordered pair, using integers or decimals. do not include the $ symbol in your answer.)

Explanation:

Step1: Recall the formula for exponential functions

The general form of an exponential function is \( C(t)=C_0(1 + r)^t\), where \( C_0\) is the initial value (when \( t = 0\)), \( r\) is the growth rate, and \( t\) is the number of years.
In the given function \( C(t)=9.2(1.038)^t\), when \( t = 0\), we have:

Step2: Calculate \( C(0)\)

Substitute \( t = 0\) into the function \( C(t)=9.2(1.038)^t\).
Using the property \( a^0=1\) for \( a
eq0\), we get \( C(0)=9.2\times(1.038)^0\).
Since \( (1.038)^0 = 1\), then \( C(0)=9.2\). The ordered - pair is \((t,C(t))=(0,9.2)\).

Step3: Interpret the \( C -\)intercept

The \( t\) - value represents the number of years after \( 2014\). When \( t = 0\), it corresponds to the year \( 2014\). The function \( C(t)\) represents personnel expenses in millions of dollars. So, the \( C -\)intercept \( C(0) = 9.2\) means that the personnel expenses in \( 2014\) were \( 9.2\) million dollars.

Answer:

The \( C -\)intercept is \( 9.2\). The ordered pair is \((0,9.2)\). The practical meaning of the \( C -\)intercept is the personnel expenses (in millions of dollars) in the year \( 2014\) (when \( t = 0\)).