Question 7 of 12intermediate🔧 ApplyNumerical4 marks

A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Correct Answer

The height of the pedestal is 0.8(3\sqrt{3} + 1) m or approximately 2.19 m.

Exercise: EXERCISE 9.1 | Q: 8 | (Page Page 10)
For More Understanding

Explanation

This problem involves two right triangles sharing the same base. The pedestal and statue together form a vertical structure where we need to find the pedestal height using trigonometric ratios. The key insight is that both triangles share the same distance from the observation point to the pedestal.

Solution Steps

  1. Step 1: Let the height of the pedestal be h metres and the distance from the observation point to the pedestal be x metres.

  2. Step 2: For the pedestal with angle of elevation 45°: tan 45° = h/x. Since tan 45° = 1, we get h = x.

  3. Step 3: For the top of the statue with angle of elevation 60°: tan 60° = (h + 1.6)/x. Since tan 60° = 3\sqrt{3}, we get 3\sqrt{3} = (h + 1.6)/x.

  4. Step 4: Substituting x = h from Step 2: 3\sqrt{3} = (h + 1.6)/h. This gives 3\sqrt{3} = 1 + 1.6/h.

  5. Step 5: Solving for h: 3\sqrt{3} - 1 = 1.6/h, therefore h = 1.6/(3\sqrt{3} - 1).

  6. Step 6: Rationalising the denominator: h = 1.6(3\sqrt{3} + 1)/((3\sqrt{3} - 1)(3\sqrt{3} + 1)) = 1.6(3\sqrt{3} + 1)/(3 - 1) = 0.8(3\sqrt{3} + 1) m.

  7. Step 7: Taking 3\sqrt{3} ≈\approx 1.732: h ≈\approx 0.8×2.7320.8 \times 2.732 ≈\approx 2.19 m.

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