1. **Problem Statement:** Find the length of the line segment NO where N = (4,0) and O = (12,0) in a square space of dimensions $2a \times 2a \times 2a$. 2. **Formula:** Length between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by: $$\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculation:** $$\text{Length NO} = \sqrt{(12 - 4)^2 + (0 - 0)^2} = \sqrt{8^2} = 8$$ 4. **Answer:** Length of NO is $8$ units. --- 5. **Problem Statement:** Calculate the length related to a triangle with an angle of $110^\circ$, height falling at midpoint 3.5 units, and meeting at apex 4.2 units. 6. **Explanation:** Using trigonometry and triangle properties, the length can be found by applying the Law of Cosines or height relations. 7. **Calculation:** Assuming the base is split into two equal parts of 3.5 units, and height is 4.2 units, the length of the side adjacent to the angle is: $$\text{Length} = \sqrt{3.5^2 + 4.2^2} = \sqrt{12.25 + 17.64} = \sqrt{29.89} \approx 5.47$$ 8. **Answer:** Length is approximately $5.47$ units. --- 9. **Problem Statement:** How many base squares are obtainable with total length 30.0 units? 10. **Explanation:** If each base square has side length $s$, number of squares $n = \frac{30.0}{s}$. 11. **Answer:** Depends on the side length $s$ of each square; if $s$ is known, calculate $n$ accordingly. --- 12. **Problem Statement:** If the perimeter is 54 units, find the base length of the shape. 13. **Explanation:** Perimeter $P = 2 \times (\text{base} + \text{height})$ for rectangle. 14. **Calculation:** Assuming height known or base is $b$, then: $$54 = 2(b + h) \Rightarrow b = \frac{54}{2} - h = 27 - h$$ 15. **Answer:** Base length depends on height $h$; if $h$ is given, substitute to find $b$. --- 16. **Problem Statement:** Find radius $r$ of the big circle inside the triangle. 17. **Explanation:** Radius of inscribed circle $r = \frac{A}{s}$ where $A$ is area and $s$ is semiperimeter. 18. **Answer:** Calculate area and semiperimeter to find $r$. --- 19. **Problem Statement:** Find base of prism when height is 19 cm using Pythagoras theorem. 20. **Formula:** $$a^2 + b^2 = c^2$$ 21. **Answer:** Base length can be found if hypotenuse or other side is known. --- 22. **Problem Statement:** Find volume of sphere with radius 12 cm. 23. **Formula:** $$V = \frac{4}{3} \pi r^3$$ 24. **Calculation:** $$V = \frac{4}{3} \pi (12)^3 = \frac{4}{3} \pi 1728 = 2304\pi \approx 7238.23$$ 25. **Answer:** Volume is approximately $7238.23$ cubic cm. --- 26. **Problem Statement:** Calculate volume of solid shape with dimensions 10 m length, 6 m height, 9 m visible length, and 4 m width. 27. **Formula:** Volume $V = \text{length} \times \text{width} \times \text{height}$ 28. **Calculation:** $$V = 10 \times 4 \times 6 = 240$$ cubic meters 29. **Answer:** Volume is $240$ cubic meters. --- 30. **Problem Statement:** Find volume of rectangular polygon with base lengths 25 m, 50 m and heights 15 m, 55 m. 31. **Explanation:** Volume can be approximated by multiplying base area and height. 32. **Answer:** Depends on shape specifics; if prism, calculate accordingly. --- 33. **Problem Statement:** Find volume of dome with radius 19 m and height 15 m. 34. **Formula:** Volume of spherical cap: $$V = \frac{1}{3} \pi h^2 (3r - h)$$ 35. **Calculation:** $$V = \frac{1}{3} \pi (15)^2 (3 \times 19 - 15) = \frac{1}{3} \pi 225 (57 - 15) = \frac{1}{3} \pi 225 \times 42 = 3150\pi \approx 9896.06$$ 36. **Answer:** Volume is approximately $9896.06$ cubic meters.