UPSC CDS 1 2020 Math Paper

Concept:Use the property of equal slopes for collinear points to set up a proportion.Explanation:Let the shorter tower be at $A$ with height $40$ ft and the taller tower at $C$ with height $H$ ft.Let the pole be at $P$ on the extension of $AC$ beyond $A$, with $AP = 75$ ft and $AC = 100$ ft.So the distance from the pole to the taller tower is $PC = 75 + 100 = 175$ ft.The tips of the towers and the pole are collinear.The vertical drop from the shorter tower tip to the pole tip is $40 - 6 = 34$ ft over a horizontal distance of $75$ ft.The vertical rise from the pole tip to the taller tower tip is $H - 6$ ft over a horizontal distance of $175$ ft.Since the points are on the same line, the slopes are equal:$\frac{34}{75} = \frac{H-6}{175}$$H - 6 = \frac{34}{75} \times 175 = 34 \times \frac{7}{3} = \frac{238}{3} \approx 79.33$$H = 79.33 + 6 = 85.33 \approx 85$ feet.Answer:$85$ feet (Option A).