Question

two less than three times the width of a rectangle is equal to the length. the area of the rectangle is 65 square ft. what is the length of the rectangle? a 9 feet b 11 feet c 13 feet d 20 feet

Explanation:

Step1: Define variables

Let the width of the rectangle be \( w \) feet. Then the length \( l = 3w - 2 \) feet (since two less than three times the width is the length). The area of a rectangle is \( A = l \times w \), and we know \( A = 65 \) square feet. So we substitute \( l \) into the area formula: \( (3w - 2) \times w = 65 \).

Step2: Solve the equation

Expand the left side: \( 3w^2 - 2w = 65 \). Rearrange to standard quadratic form: \( 3w^2 - 2w - 65 = 0 \). Now we factor the quadratic (or use quadratic formula). Let's try factoring: we need two numbers that multiply to \( 3 \times (-65) = -195 \) and add to \( -2 \). The numbers are \( -15 \) and \( 13 \). So we rewrite the middle term: \( 3w^2 - 15w + 13w - 65 = 0 \). Group and factor: \( 3w(w - 5) + 13(w - 5) = 0 \), so \( (3w + 13)(w - 5) = 0 \). Setting each factor to zero: \( 3w + 13 = 0 \) (gives negative width, discard) or \( w - 5 = 0 \), so \( w = 5 \) feet.

Step3: Find the length

Now substitute \( w = 5 \) into the length formula: \( l = 3(5) - 2 = 15 - 2 = 13 \) feet. We can check the area: \( 5 \times 13 = 65 \), which matches.

Answer:

C. 13 feet