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To solve this problem, we can use the equation of a hyperbola in its standard form and then apply the given information to find the area of the base of the cooling tower. The equation of a hyperbola in its standard form is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] where \(a\) and \(b\) are the distances from the center to the vertices along the x-axis and y-axis, respectively. Given that the distance between the two vertices of the hyperbola is 20m, we have: \[2a = 20 \Rightarrow a = 10\] We also know that the length of the real axis (transverse axis) of the hyperbola is 16m. Since \(2a\) represents this length, we have: \[2a = 16 \Rightarrow a = 8\] Now, we can find \(b\) using \(c^2 = a^2 + b^2\), where \(c\) is half of the distance between two focii. Given that half of this distance is \(\frac{2}{3}\) times half of the distance from center to base, we have: \[c = \frac{2}{3} \times \frac{125}{2} = \frac{125}{3}\] Using \(c^2 = a^2 + b^2\), we get: \[\left(\frac{125}{3}\right)^2 = 8^2 + b^2\] Solving for \(b\), we find: \[b = \sqrt{\left(\frac{125}{3}\right)^2 - 8^2} = \sqrt{\frac{15625}{9} - 64} = \sqrt{\frac{15625 - 576}{9}} = \sqrt{\frac{15049}{9}} = \frac{\sqrt{15049}}{3}\] The area (\(A_b\)) of the base of the cooling tower can be calculated using: \[A_b = \pi ab\] Substituting in our values for \(a\) and \(b\), we get: \[A_b = \pi \times 10\times \frac{\sqrt{15049}}{3}=\pi\times10\times\frac{\sqrt{15049}}{3}=150.79644737231007 m^{2}\] So, diện tích đáy của tháp là: $\boxed {150.79644737231007 m^{2}}$