Final answer:
To find the length and width of the rug with a diagonal of 26 ft and the length being 4 ft more than twice the width, we use the Pythagorean theorem to set up an equation and solve it, finding the width to be 6 ft and the length to be 16 ft. The correct option is D: Length: 16 ft, Width: 5 ft.
Step-by-step explanation:
The given problem involves the diagonal of a rectangular rug measuring 26 ft, and the length being 4 ft more than twice the width. To solve for the length and width, we can set up an equation using the Pythagorean theorem since the diagonal forms a right triangle with the length and width of the rug.
Step 1: Setting up the Equation
Let w be the width of the rug. Then the length would be 2w + 4 ft. By the Pythagorean theorem, which states that in a right triangle the square of the length of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides (length and width in this case), we can write the equation:
w^2 + (2w + 4)^2 = 26^2
Step 2: Solving the Equation
Simplifying and solving this quadratic equation will give us the width, and from there we can find the length.
w^2 + 4w^2 + 16w + 16 = 676
5w^2 + 16w - 660 = 0
Factoring or using the quadratic formula, we find that w = 6 (discarding the negative solution). Therefore, the length is 2(6) + 4 = 16 ft.
Step 3: Checking the Options
Comparing the results with the options given, we find that Option D is correct: Length: 16 ft, Width: 5 ft.