A phone company compares two rural mast designs, each stabilised by a diagonal support wire meeting the ground at the angle shown. Design Mast height (m) Wi...
Question 1 Report
A phone company compares two rural mast designs, each stabilised by a diagonal support wire meeting the ground at the angle shown.
| Design | Mast height (m) | Wire angle to ground |
|---|---|---|
| A | 18 | \(50^{\circ}\) |
| B | 15 | \(70^{\circ}\) |
- Work out the length of Design A's support wire, correct to 3 s.f. (2)
- Work out the horizontal distance from Design A's mast base to the wire's ground point, correct to 3 s.f. (2)
- Work out the length of Design B's support wire, correct to 3 s.f. (2)
- Each mast base allows only a 10 m clearance radius. State which design fits, with a reason. (2)
- Give one further advantage Design B has over Design A. (1)
Answer Details
Each mast, its wire, and the ground form a right-angled triangle with the mast as the vertical side, so the wire (the hypotenuse) is found with the sine ratio and the horizontal ground distance (adjacent to the wire's angle) with the tangent ratio.
- \[ \frac{18}{\sin50^\circ} = 23.5 \text{ m (3 s.f.)} \] [2 marks]
- \[ \frac{18}{\tan50^\circ} = 15.1 \text{ m (3 s.f.)} \] [2 marks]
- \[ \frac{15}{\sin70^\circ} = 16.0 \text{ m (3 s.f.)} \] [2 marks]
- Design B's horizontal distance is \[ \frac{15}{\tan70^\circ} = 5.46 \text{ m} \] which is within the \(10\) m clearance, while Design A's horizontal distance, \(15.1\) m from part (b), exceeds it; only Design B satisfies the constraint. [2 marks]
- Design B also uses a shorter support wire, \(16.0\) m compared with \(23.5\) m for Design A, so it needs less wire material. [1 mark]
The steeper wire angle in Design B (\(70^\circ\) against \(50^\circ\)) pulls the ground anchor much closer to the mast base, which is exactly why a steeper wire angle helps a mast fit into a tight clearance radius even though the mast itself is shorter.
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