Question
Find the value of $c$ in $[-5,-2]$ such that $f(c)$ equals the average value of $f(t)=-6t-2$ over the interval $[-5,-2]$. Enter your answer as an exact fraction if necessary.

Step 1 ：The average value of a function $f(t)$ over an interval $[a,b]$ is given by the formula $\frac{1}{b-a}{\int }_a^{b}f(t)dt$. We need to find the value of $c$ in $[-5,-2]$ such that $f(c)$ equals this average value.

Step 2 ：First, we calculate the average value of $f(t)$ over the interval $[-5,-2]$.

Step 3 ：The average value of the function over the interval $[-5,-2]$ is found to be 19.

Step 4 ：Next, we solve the equation $f(c)=19$ to find the value of $c$ that makes this true.

Step 5 ：The solution to the equation is $c=-\frac{7}{2}$.

Step 6 ：Final Answer: The value of $c$ in $[-5,-2]$ such that $f(c)$ equals the average value of $f(t)$ over the interval $[-5,-2]$ is $\boxed{{\displaystyle -\frac{7}{2}}}$.