Answer:

The approximate length of the diagonal of the initial rectangle is 79.56 cm.

Step-by-step explanation:

Let's assume the length of the rectangular sheet of paper is L cm and the breadth is B cm.

Given that the area of the rectangular sheet is 4672 sq. cm, we have the equation:

L * B = 4672

We are also given that a square is separated from the rectangular sheet, and the remainder is a rectangle of area 576 sq. cm. The area of the square is equal to the area of the rectangle, so we have:

L * B - side length * side length = 576

Since the side length of the square is equal to the length of the rectangle, we can rewrite the equation as:

L * B - L * L = 576

Simplifying:

L * (B - L) = 576

Now we need to solve this equation to find the values of L and B. However, since we only need to find the approximate length of the diagonal, we can use the given answer choices to estimate the value.

Let's calculate the approximate length of the diagonal for each answer choice:

(a) 101.63 cm:

Diagonal = sqrt(L^2 + B^2) ≈ sqrt(101.63^2 + 101.63^2) ≈ 143.71 cm

(b) 79.56 cm:

Diagonal = sqrt(L^2 + B^2) ≈ sqrt(79.56^2 + 79.56^2) ≈ 112.50 cm

(c) 121.63 cm:

Diagonal = sqrt(L^2 + B^2) ≈ sqrt(121.63^2 + 121.63^2) ≈ 171.63 cm

(d) 97.08 cm:

Diagonal = sqrt(L^2 + B^2) ≈ sqrt(97.08^2 + 97.08^2) ≈ 137.19 cm

Comparing the approximate lengths of the diagonal with the given answer choices, we can see that the closest estimate is (b) 79.56 cm.

Therefore, the approximate length of the diagonal of the initial rectangle is 79.56 cm.