To find the base side length of the triangular pyramid, we first need to calculate its volume using the mass and density. The formula for density is:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Rearranging this formula to find the volume gives:
\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{400 , \text{g}}{16 , \text{g/cm}^3} = 25 , \text{cm}^3 \]
Now, the volume \( V \) of a triangular pyramid (also known as a tetrahedron) can be calculated using the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Since the base is an equilateral triangle, we can express the base area in terms of the side length \( s \):
\[ \text{Base Area} = \frac{\sqrt{3}}{4} s^2 \]
Thus, we can equate the two volume formulas:
\[ 25 = \frac{1}{3} \times \frac{\sqrt{3}}{4} s^2 \times 5 \]
Now, simplifying the equation:
\[ 25 = \frac{5\sqrt{3}}{12} s^2 \]
Multiplying both sides by \( \frac{12}{5\sqrt{3}} \):
\[ s^2 = \frac{25 \times 12}{5 \sqrt{3}} = \frac{300}{5\sqrt{3}} = \frac{60}{\sqrt{3}} = 20\sqrt{3} \]
Taking the square root of both sides:
\[ s = \sqrt{20\sqrt{3}} = \sqrt{20} \times \sqrt{\sqrt{3}} = 2\sqrt{5} \times 3^{1/4} \]
To find a numerical approximation, we calculate:
\[ 20\sqrt{3} \approx 20 \times 1.732 = 34.64 \]
Finding the approximate value of \( s \):
\[ s = \sqrt{34.64} \approx 5.89 , \text{cm} \]
Therefore, the correct side length of the base of the pyramid is:
\[ \boxed{5.89 , \text{cm}} \]