Final Answer:
The width of the rectangle is found to be 6 feet using the given area and length values in the formula for the area of a rectangle. Therefore the correct option is b.
Step-by-step explanation:
To find the width of the rectangle, we'll use the formula for the area of a rectangle: Area = Length × Width. Given that the area of the backyard is (11 + 13√2) square feet and the length is (3 + √2) feet, we can plug these values into the formula. Let the width be represented by 'w'.
Area = Length × Width
(11 + 13√2) = (3 + √2) × w
To solve for 'w', divide both sides of the equation by (3 + √2):
w = (11 + 13√2) ÷ (3 + √2)
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator: (3 - √2).
w = [(11 + 13√2) × (3 - √2)] ÷ [(3 + √2) × (3 - √2)]
w = [33 - 11√2 + 39√2 - 13 × 2] ÷ [9 - 2]
w = (33 + 28√2 - 26) ÷ 7
w = (7 + 28√2) ÷ 7
w = 1 + 4√2
Therefore, the width of the rectangle is 6 feet (1 + 4√2). Therefore the correct option is b.