The function f is defined by f x = x + 15 5 , and f

Choice C is correct. It's given that $f\left(x\right)=\frac{x+15}{5}$ and $f\left(a\right)=10$, where $a$ is a constant. Therefore, for the given function $f$, when $x=a$, $f\left(x\right)=10$. Substituting $a$ for $x$ and $10$ for $f\left(x\right)$ in the given function $f$ yields $10=\frac{a+15}{5}$. Multiplying both sides of this equation by $5$ yields $50=a+15$. Subtracting $15$ from both sides of this equation yields $35=a$. Therefore, the value of $a$ is $35$.

Choice A is incorrect. This is the value of $a$ if $f\left(a\right)=4$.

Choice B is incorrect. This is the value of $a$ if $f\left(a\right)=5$.

Choice D is incorrect. This is the value of $a$ if $f\left(a\right)=16$.