Two pillars are placed vertically 8 feet apart. The height difference of the two pillars is 6 feet. The two ends of a rope of length 15 feet are tied to the tips of the two pillars. The portion of the length of the taller pillar that can be brought in contact with the rope without detaching the rope from the pillars is
less than 6 feet
more than 6 feet but less than 7 feet
more than 7 feet but less than 8 feet
more than 8 feet
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC CAPF – 2018
Let the tip of the shorter pillar be A and the tip of the taller pillar be B. Let C be the point on the taller pillar where the rope segment from A first touches the pillar, and the segment from C to B is along the pillar. We are looking for the length of the segment CB, let’s call it $x$.
The coordinates can be set up as A at $(0, h_1)$ and B at $(8, h_2)$. C is on the taller pillar at $(8, y_C)$, where $y_C = h_2 – x$.
The total rope length is the sum of the length of the segment AC and the segment CB.
Length of AC = $\sqrt{(8-0)^2 + (y_C – h_1)^2} = \sqrt{64 + (h_2 – x – h_1)^2}$.
Since $h_2 – h_1 = 6$, this is $\sqrt{64 + (6 – x)^2}$.
Length of CB = $h_2 – y_C = x$.
Total rope length = $\sqrt{64 + (6 – x)^2} + x = 15$.
Rearranging the equation: $\sqrt{64 + (6 – x)^2} = 15 – x$.
Squaring both sides: $64 + (6 – x)^2 = (15 – x)^2$
$64 + 36 – 12x + x^2 = 225 – 30x + x^2$
$100 – 12x = 225 – 30x$
$30x – 12x = 225 – 100$
$18x = 125$
$x = 125 / 18$.
Calculating the value: $125 \div 18 \approx 6.944$ feet.
This value is greater than 6 and less than 7.