Question

a gardener has 88 yd of fencing and wants to fence off a rectangular enclosure that borders a wall. the gardener needs no fence along the wall. what are the dimensions of the enclosure that has the largest area? answer attempt 1 out of 2

Explanation:

Step1: Define variables

Let $x$ = length of side perpendicular to wall, $y$ = length parallel to wall.
Total fencing: $2x + y = 88$, so $y = 88 - 2x$.

Step2: Set up area function

Area $A(x) = x \cdot y = x(88 - 2x) = -2x^2 + 88x$.

Step3: Find vertex of parabola

For $ax^2+bx+c$, vertex at $x = -\frac{b}{2a}$.
Here $a=-2$, $b=88$, so $x = -\frac{88}{2(-2)} = 22$.

Step4: Calculate $y$

Substitute $x=22$ into $y=88-2x$:
$y = 88 - 2(22) = 44$.

Answer:

The dimensions are 22 yards (perpendicular to the wall) and 44 yards (parallel to the wall), which gives the maximum area.