Final answer:
The dimensions of the rectangular room with a diagonal of 26 ft and one wall 14 ft longer than the other can be found using the Pythagorean theorem. The shorter wall of the room is 10 ft.
Step-by-step explanation:
To find the dimensions of a rectangular room where the diagonal is 26 ft long, and one wall is 14 ft longer than the adjacent wall, we can use the Pythagorean theorem. Let's denote the length of the shorter wall as x feet. Consequently, the longer wall will be x + 14 ft. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. For our rectangle, the diagonal is the hypotenuse, and the walls are the other two sides.
Thus, our equation is:
x2 + (x + 14)2 = 262
Expanding and simplifying gives us:
x2 + x2 + 28x + 196 = 676
Combining like terms:
2x2 + 28x + 196 = 676
2x2 + 28x - 480 = 0
Dividing the entire equation by 2 gives us a simpler quadratic equation:
x2 + 14x - 240 = 0
This can be factored into:
(x + 24)(x - 10) = 0
Hence, x = -24 or x = 10. Since the length of a wall cannot be negative, the shorter wall must be 10 ft long.
Therefore, the dimensions of the room are 10 ft by 24 ft (since the longer wall is 14 ft longer than the shorter one).