Fenced Garden Against a Barn
A gardener is building a rectangular vegetable pen against the straight side of a barn. The barn forms one entire side of the pen, so only three sides need fencing. She has 60 feet of fencing and will use all of it. Let $x$ be the length of each of the two sides perpendicular to the barn, in feet. The side parallel to the barn then has length $60 - 2x$. She wants to choose $x$ to control the area, and she is considering several area targets. The gardener also wants to know whether a neighbor's claim about a larger pen is possible with the same fencing.
1. Write the area $A$ as a function of $x$ and find the dimensions that maximize the area. What is the maximum area?
Read the worked answer
The area is
$A(x) = x(60 - 2x) = -2x^2 + 60x$.
This is a downward-opening parabola. The vertex occurs at
$x = -\frac{b}{2a} = -\frac{60}{2(-2)} = 15$.
The side parallel to the barn is $60 - 2(15) = 30$ feet. The maximum area is
$A(15) = 15(30) = 450$ square feet.
The dimensions are 15 feet by 30 feet, giving 450 square feet.
2. The gardener wants the pen to have an area of at least 400 square feet. Solve the inequality and give the corresponding range of $x$.
Read the worked answer
Set up
$-2x^2 + 60x \ge 400$.
Divide by $-2$ and reverse the inequality:
$x^2 - 30x + 200 \le 0$.
Factor:
$(x - 10)(x - 20) \le 0$.
The product is at most zero between the roots, so
$10 \le x \le 20$.
The side parallel to the barn is $60 - 2x$, which ranges from 40 feet at $x = 10$ to 20 feet at $x = 20$. So the pen has at least 400 square feet when each perpendicular side is between 10 and 20 feet.
3. A neighbor claims a 480-square-foot pen is possible with the same 60 feet of fencing. Use a discriminant to test the claim.
Read the worked answer
Set the area equal to 480:
$-2x^2 + 60x = 480$.
Divide by $-2$:
$x^2 - 30x + 240 = 0$.
The discriminant is
$b^2 - 4ac = (-30)^2 - 4(1)(240) = 900 - 960 = -60$.
Because the discriminant is negative, there is no real solution. A 480-square-foot pen is impossible with 60 feet of fencing in this barn-side configuration; the maximum is 450 square feet.