Numerical: heights and distances MCQ - Practice Questions with Answers

Let the pole height be h. At 45°, the shadow is h because tan 45° = 1. At 30°, the shadow is h/tan 30° = h√3. The increase is h√3 - h = h(√3 - 1), and this equals 80. Hence h = 80/(√3 - 1) = 40(√3 + 1) m. The expression 40(√3 - 1) reverses the rationalisation, 30(√3 + 1) uses the wrong difference, and 80(√3 + 1) doubles the actual height.

Let the height of the tower be h, and let the two angles of elevation be theta and 90° - theta. From the two points, tan theta = h/p and tan(90° - theta) = h/q. Since tan(90° - theta) equals cot theta, the product of the two tangents is 1. Therefore, (h/p) × (h/q) = 1, giving h² = pq and h = sqrt(pq). The expression p.q gives an area-like product, the cube-root expression is unrelated, and p sqrt(q) is dimensionally inconsistent for a height.

The angle of depression to the bottom of the tower is 60°, so if the horizontal distance is d, then tan 60° = 90/d and d = 30√3. The angle of depression to the top is 30°, so the vertical difference between the cliff top and the tower top is d tan 30° = 30 m. Therefore the tower height is 90 - 30 = 60 m. The values 30 m, 40 m and 70 m do not satisfy both depression-angle conditions together.