sat-dog

Choice C is correct. For a function of the form $f\left(x\right)=a{\left(r\right)}^{\frac{x}{k}}$, where $a$, $r$, and $k$ are constants and $r<1$, the value of $f\left(x\right)$ decreases by $100\left(1-r\right)\%$ for every increase of $x$ by $k$. In the given function, $a=\mathrm{5,470}$, $r=0.64$, and $k=12$. Therefore, for the given function, the value of $f\left(x\right)$ decreases by $100\left(1-0.64\right)\%$, or $36\%$, for every increase of $x$ by $12$. Since $f\left(x\right)$ represents the value, in dollars, of the equipment after $x$ months of use, it follows that the value of the equipment decreases every $12$ months by $36\%$ of its value the preceding $12$ months. Since there are $12$ months in a year, the value of the equipment decreases each year by $36\%$ of its value the preceding year. Thus, the value of $p$ is $36$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.