Question

A flagstaff 7.32 m long is at the top of a 10 m tall building. Rope $l_1$ (to building top) makes 30°. Rope $l_2$ (to flagstaff top) makes $\theta$.
(i) Find x.
(ii) Find $\theta$.
(iii) (a) Total length of ropes? OR
(iii) (b) Which rope is longer and by how much?

Hint:

Whenever the horizontal distance (base) equals the vertical distance (height), the angle of elevation is always $45^\circ$.

Answer 1

Step 1: Understanding the Concept:
This is a heights and distances problem involving two right-angled triangles with a common base $x$.
Step 2: Key Formula or Approach:
1. $\tan \alpha = \text{Opposite}/\text{Adjacent}$.
2. Pythagoras: $l = \sqrt{x^2 + h^2}$ or $l = h/\sin \alpha$.
Step 3: Detailed Explanation:
1. (i) Find x:
- In $\triangle$ with building: $\tan 30^\circ = \frac{10}{x}$.
- $\frac{1}{1.732} = \frac{10}{x} \implies x = 17.32$ m.
2. (ii) Find $\theta$:
- Total height $H = 10 + 7.32 = 17.32$ m.
- $\tan \theta = \frac{H}{x} = \frac{17.32}{17.32} = 1$.
- $\tan \theta = 1 \implies \theta = 45^\circ$.
3. (iii) (a) Total length:
- $l_1 = 10/\sin 30^\circ = 10/0.5 = 20$ m.
- $l_2 = 17.32/\sin 45^\circ = 17.32 \times \sqrt{2} = 17.32 \times 1.4 = 24.248$ m.
- Total = $20 + 24.248 = 44.248$ m.

4. (iii) (b) OR Difference:
- $l_2$ is longer.
- Difference = $24.248 - 20 = 4.248$ m.
Step 4: Final Answer:
(i) 17.32 m. (ii) 45°. (iii)(a) 44.248 m or (iii)(b) $l_2$ is longer by 4.248 m.