1. **State the problem:** Calculate the value of $$C = 2 \sqrt{\frac{81}{16} - \frac{5}{3} \left(\frac{1}{4} - \frac{1}{6} \times 0.3 \right)^2 + \left( \frac{5}{8} + 1 \times \left(1 - \frac{1}{5} \right) \right)^2}$$. 2. **Simplify inside the parentheses:** - Calculate $$\frac{1}{6} \times 0.3 = 0.05$$. - Then $$\frac{1}{4} - 0.05 = 0.25 - 0.05 = 0.2$$. 3. **Square the result:** $$0.2^2 = 0.04$$. 4. **Calculate the second bracket:** - Calculate $$1 - \frac{1}{5} = 1 - 0.2 = 0.8$$. - Multiply by 1: $$1 \times 0.8 = 0.8$$. - Add $$\frac{5}{8} = 0.625$$. - So, $$0.625 + 0.8 = 1.425$$. 5. **Square the sum:** $$1.425^2 = 2.030625$$. 6. **Substitute back into the expression under the square root:** $$\frac{81}{16} = 5.0625$$. 7. **Calculate the term with the squared difference:** $$\frac{5}{3} \times 0.04 = \frac{5}{3} \times 0.04 = 0.066666...$$. 8. **Put all together under the square root:** $$5.0625 - 0.066666... + 2.030625 = 5.0625 - 0.066666... + 2.030625 = 7.026458...$$. 9. **Calculate the square root:** $$\sqrt{7.026458...} \approx 2.6508$$. 10. **Multiply by 2:** $$C = 2 \times 2.6508 = 5.3016$$. **Final answer:** $$C \approx 5.3016$$.