1. **State the problem:** We need to find the least possible height of the post AB such that the zip wire AC makes an angle $x \geq 5^\circ$ with the horizontal. 2. **Given:** - Height of post CD = 2.6 m - Horizontal distance BD = 12 m - Angle $x$ between zip wire AC and horizontal at A satisfies $x \geq 5^\circ$ 3. **Understand the triangle:** - Points B and D are on the ground. - Post AB is vertical, height unknown. - Post CD is vertical, height 2.6 m. - Horizontal distance BD = 12 m. - The zip wire AC forms a right triangle with points A, B, and C. 4. **Calculate length CB:** Using Pythagoras theorem in triangle CBD, $$CB = \sqrt{BD^2 + CD^2} = \sqrt{12^2 + 2.6^2} = \sqrt{144 + 6.76} = \sqrt{150.76} \approx 12.28\,m$$ 5. **Relate angle $x$ to heights:** The angle $x$ is between AC and horizontal AB. Since AB is vertical and AC is the zip wire, the angle $x$ is the angle of elevation from A to C. 6. **Express $\tan x$ in terms of heights:** Let height of AB be $h$. Then vertical difference between A and C is $h - 2.6$ (since C is 2.6 m above ground). Horizontal distance BC = 12 m. So, $$\tan x = \frac{h - 2.6}{12}$$ 7. **Use the minimum angle condition:** Since $x \geq 5^\circ$, minimum $x = 5^\circ$. Calculate $\tan 5^\circ$: $$\tan 5^\circ \approx 0.08749$$ Set up inequality: $$\frac{h - 2.6}{12} \geq 0.08749$$ Multiply both sides by 12: $$h - 2.6 \geq 12 \times 0.08749 = 1.0499$$ Add 2.6 to both sides: $$h \geq 1.0499 + 2.6 = 3.6499$$ 8. **Final answer:** The least possible height of post AB is approximately $$h = 3.65\,m$$ (correct to 3 significant figures).