The Amazing Problem
Number Theory
Level
5
Someone who is 5 feet tall is looking, upwards at an angle of 40 degrees, at a painting. He steps back two feet to get a better view and looks at the painting now with an upward angle of 33 degrees. How far from the painting is the ground? (Answer in feet, to 1 decimal place.)
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The answer is 10.7.
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1 solution
P is the location of the painting (the bottom edge, we presume); p is the distance from P to the 5-ft level of the observer's head; x is the distance of the observer at the first view; x+2 is the distance at the second view.
We see that tan(40°) = p/x, and that tan(33°) = p/(x+2). Rewrite these as p = x·tan(40°) and p = (x+2)tan(33°). We now have two expressions for the same quantity p, so they are equal: x·tan(40°) = (x+2)tan(33°) = x·tan(33°) + 2·tan(33°). Solving for x gives x = 2·tan(33°) / (tan(40°) - tan(33°)) = 6.8 ft (approximately).
From the p = x·tan(40°) equation we get p = 5.7 ft (approx.). From this, the painting's bottom edge is 5.7 + 5 = 10.7 ft from the floor.