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 ：First, we need to calculate the average value of the function \(f(t) = -6t - 2\) over the interval \([-5,-2]\). The average value of a function over an interval \([a, b]\) is given by the formula \(\frac{1}{b-a}\int_{a}^{b}f(t)dt\).

Step 2 ：Next, we need to find the value of \(c\) in \([-5,-2]\) such that \(f(c)\) equals this average value. We do this by solving the equation \(f(c) = \text{average value}\) for \(c\).

Step 3 ：The solution to the equation \(f(c) = \text{average value}\) is \(c = -\frac{7}{2}\).

Step 4 ：This means that the value of \(c\) in \([-5,-2]\) such that \(f(c)\) equals the average value of \(f(t)\) over the interval \([-5,-2]\) is \(-\frac{7}{2}\).

Step 5 ：Final Answer: \(c = \boxed{-\frac{7}{2}}\)