Question

1. a rectangular corner lot, originally twice as long as it was wide, lost a 2 - meter strip along two adjacent sides due to street widening. its new area is 684 m². find its new dimensions.

Explanation:

Step1: Let the original width be \(x\) meters.

The original length is \(2x\) meters.

Step2: Calculate the original area formula.

The original area \(A = 2x\times x=2x^{2}\) square - meters.

Step3: Find the new dimensions.

The new width is \((x - 2)\) meters and the new length is \((2x - 2)\) meters.

Step4: Set up the equation for the new - area.

The new area is \((x - 2)(2x - 2)=684\). Expand the left - hand side: \(2x^{2}-2x-4x + 4=684\), which simplifies to \(2x^{2}-6x+4 = 684\), and further to \(2x^{2}-6x-680 = 0\). Divide through by 2: \(x^{2}-3x - 340=0\).

Step5: Solve the quadratic equation.

For the quadratic equation \(x^{2}-3x - 340 = 0\), where \(a = 1\), \(b=-3\), \(c=-340\). Use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{3\pm\sqrt{(-3)^{2}-4\times1\times(-340)}}{2\times1}=\frac{3\pm\sqrt{9 + 1360}}{2}=\frac{3\pm\sqrt{1369}}{2}=\frac{3\pm37}{2}\). We get two solutions for \(x\): \(x_1=\frac{3 + 37}{2}=20\) and \(x_2=\frac{3-37}{2}=-17\). Since the dimension cannot be negative, \(x = 20\).

Step6: Calculate the new dimensions.

The new width is \(x - 2=20 - 2 = 18\) meters and the new length is \(2x-2=2\times20 - 2=38\) meters.

Answer:

The new dimensions are 18 meters and 38 meters.