1. **Stating the problem:** We are asked to solve problem 13c, which involves the function $$c(t) = \frac{5t}{2t^2 + 7}$$ and likely requires finding its derivative or analyzing its behavior. 2. **Formula used:** To find the rate of change or derivative of a rational function $c(t) = \frac{f(t)}{g(t)}$, we use the quotient rule: $$c'(t) = \frac{f'(t)g(t) - f(t)g'(t)}{(g(t))^2}$$ where $f(t) = 5t$ and $g(t) = 2t^2 + 7$. 3. **Calculate derivatives of numerator and denominator:** $$f'(t) = 5$$ $$g'(t) = 4t$$ 4. **Apply the quotient rule:** $$c'(t) = \frac{5(2t^2 + 7) - 5t(4t)}{(2t^2 + 7)^2}$$ 5. **Simplify the numerator:** $$5(2t^2 + 7) - 5t(4t) = 10t^2 + 35 - 20t^2 = -10t^2 + 35$$ 6. **Write the derivative:** $$c'(t) = \frac{-10t^2 + 35}{(2t^2 + 7)^2}$$ 7. **Interpretation:** This derivative gives the rate of change of $c(t)$ with respect to $t$. **Final answer:** $$\boxed{c'(t) = \frac{-10t^2 + 35}{(2t^2 + 7)^2}}$$