We can use the Pythagorean theorem to find the diagonal distance across the rug. According to the theorem, the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side (the hypotenuse).
In this case, the length and width of the rug form the two shorter sides of a right triangle, and the diagonal distance is the hypotenuse. So we can use the formula:
diagonal^2 = length^2 + width^2
Plugging in the values given in the problem, we get:
diagonal^2 = 8^2 + 5^2
diagonal^2 = 64 + 25
diagonal^2 = 89
To solve for the diagonal, we take the square root of both sides:
diagonal = sqrt(89)
Using a calculator, we get:
diagonal ≈ 9.43 feet
Therefore, the diagonal distance across the rug is approximately 9.43 feet.