1. **Problem statement:** A rectangular garden is to be built using 100 meters of fencing. One side is along a wall, so fencing is needed only for the other three sides. We want to: a) Write an equation for the area in terms of $x$. b) Find the dimensions that give the maximum area. 2. **Define variables:** Let $x$ be the length of the side perpendicular to the wall. Let $y$ be the length of the side parallel to the wall (the side opposite the wall). 3. **Write the fencing constraint:** Since one side is along the wall, fencing is needed for two sides of length $x$ and one side of length $y$. So the total fencing used is: $$2x + y = 100$$ 4. **Express $y$ in terms of $x$:** $$y = 100 - 2x$$ 5. **Write the area $A$ as a function of $x$:** Area is length times width: $$A = x \times y = x(100 - 2x) = 100x - 2x^2$$ 6. **Find the maximum area:** To maximize $A$, take the derivative with respect to $x$ and set it to zero: $$\frac{dA}{dx} = 100 - 4x = 0$$ Solve for $x$: $$4x = 100 \implies x = 25$$ 7. **Find $y$ corresponding to $x=25$:** $$y = 100 - 2(25) = 100 - 50 = 50$$ 8. **Verify maximum using second derivative test:** $$\frac{d^2A}{dx^2} = -4 < 0$$ Since the second derivative is negative, the critical point at $x=25$ is a maximum. 9. **Final answer:** The dimensions that give the maximum area are: $$x = 25 \text{ meters (perpendicular sides)}$$ $$y = 50 \text{ meters (side opposite the wall)}$$ The maximum area is: $$A = 25 \times 50 = 1250 \text{ square meters}$$