9. A superball bounces to $\frac{3}{4}$ of its initial height when dropped on pavement If the ball ground for the $6^{\text {th }}$ time?

Step 1 ：Let the initial height of the ball be \(h\).

Step 2 ：The common ratio \(r = \frac{3}{4}\) and the number of terms \(n = 6\).

Step 3 ：Use the geometric series formula to find the sum of the distances traveled during each bounce: \(S_n = \frac{a_1(1 - r^n)}{1 - r}\).

Step 4 ：Calculate the total distance traveled after the 6th bounce by multiplying the sum of the geometric series by 2: \(2 \times S_n\).

Step 5 ：Plug in the values for \(a_1\), \(r\), and \(n\) into the formula and simplify: \(2 \times \frac{h(1 - (\frac{3}{4})^6)}{1 - \frac{3}{4}}\).

Step 6 ：Final Answer: The total distance traveled by the superball after it hits the ground for the 6th time is approximately \(\boxed{6.576}\).