Question

5.) the length of a rectangle is 6 inches more than its width. the area of the rectangle is 91 square inches. find the dimensions of the rectangle.

Explanation:

Step1: Define variables

Let the width of the rectangle be \( x \) inches. Then the length is \( x + 6 \) inches (since length is 6 inches more than width).

Step2: Use area formula

The area of a rectangle is given by \( A=\text{length}\times\text{width} \). We know the area \( A = 91 \) square inches. So we set up the equation:
\( x(x + 6)=91 \)

Step3: Expand and simplify equation

Expand the left - hand side: \( x^{2}+6x = 91 \)
Rearrange it to a quadratic equation in standard form: \( x^{2}+6x - 91=0 \)

Step4: Solve the quadratic equation

We can solve the quadratic equation \( x^{2}+6x - 91 = 0 \) by factoring. We need to find two numbers that multiply to \( - 91 \) and add up to 6. The numbers are 13 and - 7.
So, \( x^{2}+6x - 91=(x + 13)(x - 7)=0 \)
Setting each factor equal to zero:
\( x+13 = 0\) gives \( x=-13 \) (discard since width cannot be negative)
\( x - 7=0\) gives \( x = 7 \)

Step5: Find length and width

The width \( x = 7 \) inches.
The length is \( x + 6=7 + 6 = 13 \) inches.

Answer:

The width of the rectangle is 7 inches and the length is 13 inches.