Question
Find the value of $c$ in $[-5,-2]$ such that $f(c)$ equals the average value of $f(t)=-6 t-2$ 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 \(\boxed{-\frac{7}{2}}\).