Question

question 20 · 1 point a farmer wants to construct a fence around an area of 3456 square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. what dimensions should the fenced - in area have in order to minimize the length of fencing used? provide your answer below: length = □ ft, width = □ ft

Explanation:

Step1: Let the length of the rectangle be $l$ and the width be $w$. The area $A = lw=3456$. So, $l=\frac{3456}{w}$.

The total length of the fence $P = 3w + 2l$ (since we are dividing the rectangle in half parallel to one - side, assume the division is parallel to the width).

Step2: Substitute $l=\frac{3456}{w}$ into the perimeter formula:

$P = 3w+2\times\frac{3456}{w}=3w+\frac{6912}{w}$.

Step3: To minimize $P$, take the derivative with respect to $w$.

$P'(w)=3-\frac{6912}{w^{2}}$.
Set $P'(w) = 0$ to find the critical points.
$3-\frac{6912}{w^{2}} = 0$.
$3=\frac{6912}{w^{2}}$.
$w^{2}=\frac{6912}{3}=2304$.
$w = 48$ (we take the positive value since width cannot be negative).

Step4: Find the length.

Since $l=\frac{3456}{w}$, when $w = 48$, $l=\frac{3456}{48}=72$.

Answer:

Length = 72 ft, Width = 48 ft