Final answer:
To determine the height of the rectangle, use the Pythagorean theorem with the height as 'h' and the width as 'h + 34 inches'. Solve the quadratic equation derived from the theorem to find the height to be 16 inches.
Step-by-step explanation:
To find the height of a rectangle where the width is 34 inches longer than the height and the diagonal measures 50 inches, we can utilize the Pythagorean theorem. This theorem is used to find the lengths of the sides of a right triangle.
Let h represent the height of the rectangle. Accordingly, the width will then be h + 34 inches. Since the rectangle's diagonal forms the hypotenuse of a right triangle whose legs are the width and the height of the rectangle, we can set up the equation:
h^2 + (h + 34)^2 = 50^2
To solve for h, first expand the equation:
h^2 + h^2 + 68h + 34^2 = 50^2
Simplify the equation by combining like terms:
2h^2 + 68h + 34^2 - 50^2 = 0
This will give us a quadratic equation that we can solve to find the value of h.
After doing the calculations, we find h to be equal to 16, which implies that the rectangle's height is 16 inches.