Yusuf has 44 m of fencing to build a three-sided fence around a rectangular plot of land that sits on a riverbank. (The fourth side of the enclosure would be the river.) The area of the land is 192 square meters. List each set of possible dimensions (length and width) of the field.

Sure, let's solve this step by step.

1. Define the variables:

   • Let $L$ be the length of the rectangular plot parallel to the river.
   • Let $W$ be the width of the rectangular plot perpendicular to the river.

2. Set up the equations:

   • The perimeter of the three-sided fence is given by $2W + L = 44$ meters.
   • The area of the rectangular plot is given by $L \times W = 192$ square meters.

3. Express $L$ in terms of $W$ using the perimeter equation: $L = 44 - 2W$

4. Substitute $L$ in the area equation: $(44 - 2W) \times W = 192$

5. Simplify and solve the quadratic equation: $44W - 2W^2 = 192$ $2W^2 - 44W + 192 = 0$ $W^2 - 22W + 96 = 0$

6. Factor the quadratic equation: $(W - 16)(W - 6) = 0$

7. Solve for $W$: $W = 16 \quad \text{or} \quad W = 6$

8. Find the corresponding $L$ values:

   • For $W = 16$: $L = 44 - 2 \times 16 = 44 - 32 = 12$
   • For $W = 6$: $L = 44 - 2 \times 6 = 44 - 12 = 32$

9. List the possible dimensions:

   • The possible dimensions of the field are:
     • Length $L = 12$ meters and Width $W = 16$ meters
     • Length $L = 32$ meters and Width $W = 6$ meters

So, the sets of possible dimensions for the field are:

1. $12$ meters by $16$ meters
2. $32$ meters by $6$ meters