Question

The diagonal of a rectangular room is 13 ft long. One wall measures 7ft longer than the adjacent wall. Find the dimensions of the room.

Answer 1

Let x be the length of the shorter wall of the rectangular room and y be the length of the longer wall of the rectangular room. Since the longer wall is 7 ft longer than the shorter wall, we can write the equation y = x + 7. We can also express the diagonal of the rectangular room using the Pythagorean theorem as x^2 + y^2 = 13^2. We can solve for x and y by substituting the equation y = x + 7 into the equation x^2 + y^2 = 13^2 and then solving for x. Doing this, we get x^2 + (x + 7)^2 = 13^2. Expanding the square on the right side of the equation and then rearranging the terms, we get x^2 + 2x^2 + 14x + 49 = 169. Combining like terms, we get 3x^2 + 14x - 120 = 0. This quadratic equation can be factored as (x - 8)(3x + 15) = 0. Since the length of a side of a rectangle must be positive, we can ignore the solution x = -15/3. So, the length of the shorter wall of the rectangular room is x = 8 ft. The length of the longer wall can be found by substituting this value into the equation y = x + 7, giving us y = 8 + 7 = 15 ft. Therefore, the dimensions of the rectangular room are 8 ft by 15 ft.

Answer 2

By applying the Pythagorean theorem to the rectangular room with a given diagonal of 13 ft and one wall 7 ft longer than the other, we solve for x to find that the dimensions of the room are 5 ft by 12 ft.

To determine the dimensions of the room when the diagonal is 13 ft long and one wall is 7 ft longer than the adjacent wall, we can use the Pythagorean theorem. Let's denote the length of the shorter wall as x ft. Therefore, the length of the longer wall will be x + 7 ft. Since the diagonal forms a right triangle with the walls of the room, by the Pythagorean theorem:

c² = a² + b²

Where c is the diagonal, and a and b are the length and width of the room, respectively. Plugging in the values we have:

13² = x² + (x + 7)²

Solving for x:

169 = x² + x² + 14x + 49

2x² + 14x - 120 = 0

Dividing the entire equation by 2:

x² + 7x - 60 = 0

Factorizing the quadratic equation:

(x + 12)(x - 5) = 0

Therefore, x = -12 or x = 5. Since a dimension cannot be negative, x must be 5 ft. Hence, the shorter wall is 5 ft, and the longer wall is 5 + 7 = 12 ft. The dimensions of the room are therefore 5 ft by 12 ft.