Carrillo Goodwin
07/17/2023 · Primary School
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.
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Step-by-step Solution
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
Quick Answer
Width: 7 meters, Length: 10 meters
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