Let's break down the problem step by step.

Given:

• Height of pole AB = 10 m
• Angle of elevation of the pole AB from P = 30°
• Distance between the foot of the pole and the foot of the tower BC = 47.3 m
• A cable of length 60 m is tied from the top of the tower D to point P
• We need to find:
  1. The height of the tower CD
  2. The angle ∠APC

Step 1: Finding Distance BP

Since the angle of elevation of pole AB (10 m) from P is 30°, we use the tan formula:

$\tan 30^\circ = \frac{\text{AB}}{\text{BP}}$

$\frac{10}{BP} = \frac{1}{\sqrt{3}}$

$BP = 10 \times \sqrt{3} \approx 10 \times 1.732 = 17.32 \text{ m}$

Step 2: Finding Distance PC

Since BC (distance between foot of pole and foot of tower) is 47.3 m, and BP is 17.32 m, we can find PC:

$PC = BC + BP = 47.3 + 17.32 = 64.62 \text{ m}$

Step 3: Finding the Height of Tower (CD)

From point P, the angle of elevation to the top of the tower (D) is θ. Using the sine rule in right triangle DPC, we use:

$\sin \theta = \frac{\text{CD}}{\text{cable length (PD)}}$

Since we have PC and PD = 60 m, we use the cosine rule in ΔDPC:

$PD^2 = PC^2 + CD^2$

$60^2 = 64.62^2 + CD^2$

$3600 = 4175.34 + CD^2$

$CD^2 = 3600 - 4175.34 = 575.34$

$CD = \sqrt{575.34} \approx 23.98 \text{ m}$

So, the height of the tower CD is 23.98 m.

Step 4: Finding Angle APC

Using the sine rule:

$\sin \theta = \frac{CD}{PD} = \frac{23.98}{60} = 0.3997$

$\theta = \sin^{-1} (0.3997) \approx 23.56^\circ$

Thus, ∠APC ≈ 23.56°.

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Final Answers:

1. Height of the tower (CD) ≈ 23.98 m
2. Angle APC ≈ 23.56°

Would you like a step-by-step breakdown of any part?

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Related Questions:

1. How do you derive the sine and cosine rules for non-right triangles?
2. What happens if the cable length is increased to 70 m? How does it affect the height of the tower?
3. Can we find the angle APC using the tangent function instead?
4. How does the problem change if the angle of elevation of the tower from P is given instead?
5. What would be the maximum possible height of the tower if the cable remains 60 m long?

Tip:
In trigonometry, always check if a right triangle is involved before applying the sine and cosine rules. This simplifies calculations! 🚀