To find the length of the longest altitude of a triangle, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

Here, the area of the triangle is given as 4205 cm^2. We need to find the length of the longest altitude, which is the height of the triangle.

Let's assume the sides of the triangle are a = 35 cm, b = 54 cm, and c = 61 cm.

First, we need to find the base of the triangle. The base can be any of the three sides. Let's assume the base is a = 35 cm.

1. Finding the height using the formula:
Area = (1/2) * base * height
4205 = (1/2) * 35 * height
4205 = 17.5 * height
height = 4205 / 17.5
height ≈ 240.29 cm

2. Checking if the height we found is the longest altitude:
To determine if the height we found is the longest altitude, we need to check if it is perpendicular to the base. If it is, then it is the longest altitude.

To check if the height is perpendicular to the base, we can use the Pythagorean theorem. According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Let's calculate:

c^2 = a^2 + b^2
61^2 = 35^2 + height^2
3721 = 1225 + height^2
height^2 = 3721 - 1225
height^2 ≈ 2496
height ≈ √2496
height ≈ 49.96 cm

Since the value we found for the height (49.96 cm) is less than the value we initially found (240.29 cm), it means that the longest altitude is not the height we initially found.

3. Finding the longest altitude:
To find the longest altitude, we need to find the height that is perpendicular to the longest side of the triangle.

Let's assume the longest side is c = 61 cm. We can use the same formula as before:

Area = (1/2) * base * height
4205 = (1/2) * 61 * height
4205 = 30.5 * height
height = 4205 / 30.5
height ≈ 137.70 cm

Therefore, the length of the longest altitude of the triangle is approximately 137.70 cm, which corresponds to option C.