To find the smallest altitude of a triangle, we first need to understand what an altitude is. An altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side. The length of the altitude can be calculated using the formula:

Altitude = (2 * Area) / Base

where Area is the area of the triangle and Base is the length of the side to which the altitude is drawn.

1. Finding the Area of the Triangle:
To find the area of the triangle, we can use Heron's formula. Heron's formula states that the area of a triangle with sides a, b, and c is given by:

Area = √(s * (s - a) * (s - b) * (s - c))

where s is the semi-perimeter of the triangle, which is calculated as:

s = (a + b + c) / 2

2. Calculating the Smallest Altitude:
Now, let's calculate the area of the triangle using the given sides: 50 cm, 78 cm, and 112 cm.

s = (50 + 78 + 112) / 2 = 120 cm

Area = √(120 * (120 - 50) * (120 - 78) * (120 - 112))
= √(120 * 70 * 42 * 8)
= √(1,123,200)
≈ 1,059.68 cm²

Next, we need to determine the base of the altitude for each side of the triangle. The base will be the side opposite to the altitude we want to find.

For the side lengths 50 cm, 78 cm, and 112 cm, the bases will be 112 cm, 50 cm, and 78 cm, respectively.

Using the formula for altitude, we can calculate the smallest altitude:

Altitude = (2 * Area) / Base

For the smallest side length of 50 cm:
Altitude = (2 * 1,059.68) / 50
= 21.19 cm

For the side length of 78 cm:
Altitude = (2 * 1,059.68) / 78
= 27.12 cm

For the largest side length of 112 cm:
Altitude = (2 * 1,059.68) / 112
= 18.94 cm

So, the smallest altitude is approximately 18.94 cm, which is closest to option C) 30 cm.