Question

the length of a rectangle is 10 ft less than three times the width, and the area of the rectangle is 77 ft². find the dimensions of the rectangle. length : ft width : ft

Explanation:

Step1: Set up variables

Let the width of the rectangle be \(x\) ft. Then the length is \((3x - 10)\) ft.

Step2: Use the area formula

The area of a rectangle is \(A=\text{length}\times\text{width}\). So we have the equation \(x(3x - 10)=77\).
Expand it: \(3x^{2}-10x - 77 = 0\).

Step3: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c=0\) (\(a = 3\), \(b=-10\), \(c = - 77\)), use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-10)^{2}-4\times3\times(-77)=100 + 924=1024\).
Then \(x=\frac{10\pm\sqrt{1024}}{6}=\frac{10\pm32}{6}\).
We get two solutions for \(x\):
\(x_{1}=\frac{10 + 32}{6}=\frac{42}{6}=7\) and \(x_{2}=\frac{10-32}{6}=\frac{-22}{6}=-\frac{11}{3}\).
Since the width cannot be negative, we take \(x = 7\).

Step4: Find the length

If \(x = 7\) (width), then the length \(l=3x-10=3\times7 - 10=21 - 10 = 11\).

Answer:

Length: \(11\) ft, Width: \(7\) ft