1. The length of a rectangle is 4 meters less than twice its width. The area of the rectangle is \( 70 \mathrm{~m}^{2} \). Find the dimensions of the rectangle.

Let's denote the width of the rectangle as \( w \) meters. Given that the length of the rectangle is 4 meters less than twice its width, we can express the length as \( 2w - 4 \) meters. The area of a rectangle is given by the formula: \[ \text{Area} = \text{Length} \times \text{Width} \] Substitute the expressions for length and width into the area formula: \[ 70 = (2w - 4) \times w \] Now, we can solve this equation to find the dimensions of the rectangle. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(70=\left(2w-4\right)w\) - step1: Multiply the terms: \(70=w\left(2w-4\right)\) - step2: Swap the sides: \(w\left(2w-4\right)=70\) - step3: Expand the expression: \(2w^{2}-4w=70\) - step4: Move the expression to the left side: \(2w^{2}-4w-70=0\) - step5: Factor the expression: \(2\left(w-7\right)\left(w+5\right)=0\) - step6: Divide the terms: \(\left(w-7\right)\left(w+5\right)=0\) - step7: Separate into possible cases: \(\begin{align}&w-7=0\\&w+5=0\end{align}\) - step8: Solve the equation: \(\begin{align}&w=7\\&w=-5\end{align}\) - step9: Rewrite: \(w_{1}=-5,w_{2}=7\) The solutions to the equation are \( w = -5 \) and \( w = 7 \). Since the width of a rectangle cannot be negative, the width of the rectangle is 7 meters. Now, we can find the length of the rectangle using the expression \( 2w - 4 \): \[ \text{Length} = 2 \times 7 - 4 = 14 - 4 = 10 \text{ meters} \] Therefore, the dimensions of the rectangle are: - Width: 7 meters - Length: 10 meters