12. It took 3 hours to row a boat 18 km against the current. The return trip with the current took \( 1 \frac{1}{2} \) hours. Find the speed of the rowboat in still water.

Let's denote the speed of the rowboat in still water as \( v \) and the speed of the current as \( c \). Given: - Time taken to row against the current = 3 hours - Distance covered against the current = 18 km - Time taken to row with the current = \( 1 \frac{1}{2} \) hours - Distance covered with the current = 18 km We can use the formula: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] For the trip against the current: \[ v - c = \frac{18}{3} \] \[ v - c = 6 \] For the trip with the current: \[ v + c = \frac{18}{1.5} \] \[ v + c = 12 \] Now, we have a system of two equations: \[ v - c = 6 \] \[ v + c = 12 \] We can solve this system of equations to find the values of \( v \) and \( c \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}v-c=6\\v+c=12\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}c=-6+v\\v+c=12\end{array}\right.\) - step2: Substitute the value of \(c:\) \(v-6+v=12\) - step3: Add the terms: \(2v-6=12\) - step4: Move the constant to the right side: \(2v=12+6\) - step5: Add the numbers: \(2v=18\) - step6: Divide both sides: \(\frac{2v}{2}=\frac{18}{2}\) - step7: Divide the numbers: \(v=9\) - step8: Substitute the value of \(v:\) \(c=-6+9\) - step9: Calculate: \(c=3\) - step10: Calculate: \(\left\{ \begin{array}{l}c=3\\v=9\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}c=3\\v=9\end{array}\right.\) - step12: Rewrite: \(\left(c,v\right) = \left(3,9\right)\) The solution to the system of equations is \( c = 3 \) and \( v = 9 \). Therefore, the speed of the rowboat in still water is 9 km/h.