To find the height of an equilateral triangle, you can use the relationship between the height (h) and the side length (s). For an equilateral triangle, the height can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} \times s \]
Given that the side length \(s = 8\) meters, we need to calculate the height using the approximation \(\sqrt{3} \approx \frac{7}{4}\).
Substituting the values into the formula, we have:
\[ h = \frac{\sqrt{3}}{2} \times 8 \]
Using the approximation for \(\sqrt{3}\):
\[ h = \frac{\frac{7}{4}}{2} \times 8 \]
Now simplify the fraction:
\[ \frac{7}{4} \div 2 = \frac{7}{4} \times \frac{1}{2} = \frac{7}{8} \]
Now, plug this back into the height formula:
\[ h = \frac{7}{8} \times 8 \]
Now performing the multiplication:
\[ h = 7 \text{ meters} \]
Therefore, the approximate height of the equilateral triangle with a side length of 8 meters is \( \boxed{7} \) meters.