Final answer:
By applying the Pythagorean theorem to the given rectangle with a diagonal of 29 inches and width that is 1 inch shorter than the length, we find that the width of the rectangle is 20 inches and the length is 21 inches.
Step-by-step explanation:
To solve for the lengths of the sides of the rectangle, we can apply the Pythagorean theorem, which 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.
Here, the diagonal of the rectangle acts as the hypotenuse, and the width and length are the other two sides.
Let's denote the width of the rectangle as w inches. According to the problem, the length is 1 inch longer than the width, so it will be w + 1 inches. The diagonal is given to be 29 inches.
Therefore, applying the Pythagorean theorem:
- w2 + (w + 1)2 = 292
- w2 + w2 + 2w + 1 = 841
- 2w2 + 2w + 1 = 841
- 2w2 + 2w - 840 = 0 (simplifying the equation)
- w2 + w - 420 = 0 (dividing by 2 for easier calculation)
- (w + 21)(w - 20) = 0 (factoring the quadratic equation)
Solving for w, we get two possible solutions: w = -21 or w = 20. We discard the negative value because a length cannot be negative, leaving us with w = 20 inches as the width of the rectangle. Consequently, the length of the rectangle is w + 1 = 20 + 1 = 21 inches.