Question

the length of a rectangle is 13 centimeters less than six times its width. its area is 15 square centimeters. find the dimensions of the rectangle.

the width is □ and the length is □

Explanation:

Step1: Define variables

Let the width of the rectangle be \( x \) centimeters. Then the length is \( 6x - 13 \) centimeters (since length is 13 cm less than six times width).

Step2: Use area formula

The area of a rectangle is \( \text{length} \times \text{width} \). So we set up the equation:
\( x(6x - 13) = 15 \)

Step3: Expand and rearrange

Expand the left side: \( 6x^2 - 13x = 15 \)
Rearrange into standard quadratic form: \( 6x^2 - 13x - 15 = 0 \)

Step4: Solve quadratic equation

We can solve \( 6x^2 - 13x - 15 = 0 \) using the quadratic formula \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 6 \), \( b=-13 \), \( c = -15 \).
First, calculate the discriminant \( D = b^2 - 4ac = (-13)^2 - 4\times6\times(-15)=169 + 360 = 529 \)
Then \( x=\frac{13\pm\sqrt{529}}{12}=\frac{13\pm23}{12} \)
We have two solutions:
\( x_1=\frac{13 + 23}{12}=\frac{36}{12}=3 \)
\( x_2=\frac{13 - 23}{12}=\frac{-10}{12}=-\frac{5}{6} \)
Since width can't be negative, we take \( x = 3 \)

Step5: Find length

Substitute \( x = 3 \) into the length formula: \( 6x - 13 = 6\times3 - 13 = 18 - 13 = 5 \)

Answer:

The width is \( 3 \) centimeters and the length is \( 5 \) centimeters.