1. **Problem statement:** Given the function $q = f(t) = 54 - 2.5t$ representing the volume of water in quarts in a sink $t$ seconds after removing the stopper, find the inverse function $f^{-1}$. 2. **Formula and rules:** To find the inverse function, swap $q$ and $t$ and solve for the new $t$ (which will be $f^{-1}(q)$). 3. **Find $f^{-1}$:** Start with $$q = 54 - 2.5t$$ Swap $q$ and $t$: $$t = 54 - 2.5q$$ Solve for $q$: $$t - 54 = -2.5q$$ Divide both sides by $-2.5$: $$q = \frac{t - 54}{-2.5} = \frac{54 - t}{2.5}$$ 4. **Final inverse function:** $$f^{-1}(t) = \frac{54 - t}{2.5}$$ 5. **Interpretation:** - Input of $f^{-1}$ is the volume of water in the sink (quarts). - Output of $f^{-1}$ is the time in seconds after removing the stopper. 6. **Time to drain completely:** When the sink is empty, volume $q=0$. Use $f^{-1}$: $$f^{-1}(0) = \frac{54 - 0}{2.5} = \frac{54}{2.5} = 21.6$$ So, it takes 21.6 seconds to drain completely. 7. **Time to lose one third of water:** One third of 54 quarts is $\frac{1}{3} \times 54 = 18$ quarts. Remaining volume after losing one third is $54 - 18 = 36$ quarts. Use $f^{-1}$: $$f^{-1}(36) = \frac{54 - 36}{2.5} = \frac{18}{2.5} = 7.2$$ So, it takes 7.2 seconds to lose one third of the water. **Final answers:** - $f^{-1}(t) = \frac{54 - t}{2.5}$ - Time to drain completely: 21.6 seconds - Time to lose one third of water: 7.2 seconds