Question

a rectangle has an area of 138 square centimeters. the length is 5 more than 3 times the width. find the length and width, in centimeters, of the rectangle.
show your work here
length:
width:

Explanation:

Step1: Set up variables

Let the width of the rectangle be \(x\) centimeters. Then the length is \((3x + 5)\) centimeters.

Step2: Use the area formula

The area of a rectangle is \(A=\text{length}\times\text{width}\). So we have the equation \(x(3x + 5)=138\).
Expand it: \(3x^{2}+5x - 138=0\).

Step3: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 3\), \(b = 5\), \(c=- 138\)), use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(5)^{2}-4\times3\times(-138)=25 + 1656=1681\).
Then \(x=\frac{-5\pm\sqrt{1681}}{6}=\frac{-5\pm41}{6}\).
We get two solutions for \(x\):
\(x_{1}=\frac{-5 + 41}{6}=\frac{36}{6}=6\) and \(x_{2}=\frac{-5-41}{6}=\frac{-46}{6}\) (reject since width cannot be negative).

Step4: Find the length

If \(x = 6\) (width), then the length \(l=3x + 5=3\times6+5=18 + 5=23\).

Answer:

length: \(23\)
width: \(6\)