To maximize the area of a rectangular garden with three sides fenced, given a total fencing of 180 yards, we can set up and solve this problem as follows:

1. Define Variables:

   • Let $w$ be the width (smaller dimension) of the garden. Since this is the side needing two parts of the fencing, we will have two widths to account for.
   • Let $l$ be the length (longer dimension) of the garden. This side needs only one part of the fencing.

2. Set Up the Equation for Fencing:

   • Since two widths and one length need to be fenced, we have the equation: $2w + l = 180$
   • Rearranging this for $l$: $l = 180 - 2w$

3. Area Function:

   • The area $A$ of the rectangular garden is given by: $A = l \times w$
   • Substitute $l = 180 - 2w$ into the area formula: $A = (180 - 2w) \times w$
   • Expanding this: $A = 180w - 2w^2$

4. Maximize the Area Function:

   • This is a quadratic function in terms of $w$, given by $A = -2w^2 + 180w$, which opens downward (as the coefficient of $w^2$ is negative). The maximum area will occur at the vertex of this parabola.
   • The $w$-coordinate of the vertex for a parabola $ax^2 + bx + c$ is given by $w = -\frac{b}{2a}$.
   • Here, $a = -2$ and $b = 180$, so: $w = -\frac{180}{2 \times -2} = \frac{180}{4} = 45$
   • Substituting $w = 45$ back into the equation for $l$: $l = 180 - 2(45) = 180 - 90 = 90$

5. Conclusion:

   • The dimensions of the garden that will maximize the area are:
     • Width $w = 45$ yards
     • Length $l = 90$ yards

Thus, the garden with the largest possible area will have a width of 45 yards and a length of 90 yards.

Would you like further details on any part of the solution?

Related Questions:

1. How would the dimensions change if the rancher had a different total amount of fencing, say 200 yards?
2. What is the maximum area of the garden based on the dimensions found?
3. How would the solution differ if all four sides of the garden required fencing?
4. Can we solve this using calculus to find the maximum area? How would that approach look?
5. What if the rancher wanted a square enclosure instead?

Tip:

For maximum area problems with constrained resources, setting up an equation with a single variable using given constraints can simplify finding optimal dimensions.