Question

question 1 - 13 miranda uses 80 feet of fencing to build a rectangular dog run. one side of the rectangle is the wall of mirandas house as shown in the figure. the total area of the dog run must be 460 square feet. which equation can be solved to find x, the length in feet of the dog run? -x² + 40x = 460 -2x² + 80x = 460 -\frac{x²}{2}+80x = 460 -\frac{x²}{2}+40x = 460

Explanation:

Step1: Set up the area - equation

Let the length of the dog - run parallel to the wall be $x$ and the other side be $y$. The amount of fencing is $x + 2y=80$, so $y=\frac{80 - x}{2}$. The area $A=xy$. Substitute $y$ into the area formula: $A=x\cdot\frac{80 - x}{2}=\frac{80x - x^{2}}{2}=40x-\frac{1}{2}x^{2}$. We know that $A = 460$. So the equation is $-\frac{1}{2}x^{2}+40x = 460$. Multiply through by - 2 to get $x^{2}-80x=-920$, or $x^{2}-80x + 920 = 0$.

Step2: Use the quadratic formula

The quadratic formula for a quadratic equation $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For the equation $x^{2}-80x + 920 = 0$, $a = 1$, $b=-80$, and $c = 920$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-80)^{2}-4\times1\times920=6400 - 3680=2720$. Then $x=\frac{80\pm\sqrt{2720}}{2}=\frac{80\pm4\sqrt{170}}{2}=40\pm2\sqrt{170}$.

Answer:

$x = 40\pm2\sqrt{170}$