That's a very narrow alley

Geometry Level 4

In an alleyway with tall buildings on both sides, a ladder of length 3.9 m 3.9 \text{ m} leans from the foot of the west wall on the east wall, while a ladder of length 2.5 m 2.5 \text{ m} leans the other way across the alleyway, from the foot of the east wall on to the west wall. Looking north along the alleyway, the ladders appear to cross 1 2 7 m 1\frac{2}{7} \text{ m} above the roadway.

How wide is the alleyway in metres? Round your answer to 3 decimal places


The answer is 1.500.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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2 solutions

Crossed Ladder Formula applied,
1 d + 1 e = 7 / 9. B u t d = 2. 5 2 − x 2 , e = 3. 9 2 − x 2 . ∴ 2. 5 2 − x 2 + 3. 9 2 − x 2 = 7 9 ∗ 2. 5 2 − x 2 ∗ 3. 9 2 − x 2 S q u a r i n g b o t h s i d e t w i c e w i t h p r o p e r a d j u s t m e n t , f o r x > 0 , T h e r e s u l t i n g q u a d r a t i c i n x 2 g i v e s x = 15 \dfrac 1 d~+~\dfrac 1 e =7/9. \\ ~~~\\ But~d=\sqrt{2.5^2 - x^2},~~~~~~~e=\sqrt{3.9^2 - x^2}.\\ ~~~\\ \therefore~ \sqrt{2.5^2 - x^2}+ \sqrt{3.9^2 - x^2}\\ =\frac 7 9 * \sqrt{2.5^2 - x^2}* \sqrt{3.9^2 - x^2}\\ ~~~~~\\ Squaring ~both~side~twice~with~proper~adjustment,~~for~ x>0,\\ ~~~~\\ The ~resulting~ quadratic~ in ~x^2~gives~x=\color{#D61F06}{\Large 15 }

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Ahmad Saad
May 29, 2016

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Can you give a solution which doesn't use trial and error please? We want to make sure this is the only solution after all.

Sharky Kesa - 5 years ago

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