Chapter 4.8, Problem 30P

To determine

To find the dimensions of the corral.

Answer to Problem 30P

Width of the border be 2.5 meters.

Explanation of Solution

Given information: A rancher use 160 yd of fencing to enclose a rectangular corral and to divide it into parts by a fence parallel to the sider sides of the corral. Area of the corral is $1000{\text{ yd}}^2$ .

Calculation: The required solution is obtained as,

Let $x$ be width of corral.

  $y$ be length of corral.

Equation for the area, $xy=1000$

Equation for the amount of the fencing,

  $\begin{array}{l}3x+2y=160\\ 2y=160-3x\\ y=80-\frac{3}{2}x\end{array}$

Substitute $y=80-\frac{3}{2}x$ into the equation of area.

  $\begin{array}{l}x\left(80-\frac{3}{2}x\right)=1000\\ 80x-\frac{3}{2}{x}^2=1000\\ 160x-3x^2=2000\\ 3x^2-160x+2000=0\\ 3x^2-60x-100x+2000=0\\ 3x\left(x-20\right)-100\left(x-20\right)=0\\ \left(x-20\right)\left(3x-100\right)=0\\ x=20,100/3\end{array}$

If $x=20y=30-\frac{3}{2}x=30-\frac{3}{2}\cdot 20=50$

If $x=\frac{100}{3}y=30-\frac{3}{2}x=30-\frac{3}{2}\cdot \frac{100}{3}=30$

  $x$ must be 20, not 100/3.

Hence, the dimensions of the corral is 20yd by 50yd.