A Bird's Eye ViewMath brain teasers require computations to solve.
Jack sat in his father's office, atop a bookcase by the wide, floor-to-ceiling window. His father's employer had built this skyscraper right on the shoreline of the lake, directly across which stood the tallest tree in the state. Jack's view was spectacular, and, as he looked across the 866 ft. wide lake, he noticed something which he found interesting. When he gazed at the highest point on the tree, his line of sight was at a 30 degree incline above horizontal. When looking at the base of the tree, where it met the water at the far edge of the lake, he was looking at a 30 degree declination.
How tall is the tree?
Bonus: Later, after querying his father, he found out that the first floor of the office building was at ground level, and every subsequent floor was exactly 12 ft. above the one below. What floor is his father's office on?
HintDraw a triangle from Jack's eyes, to both the base and the peak of the tree. Make a second triangle from the tree base to the bottom of the building, then up to Jack again.
AnswerThe tree is 1000 ft. tall.
Start by picturing a triangle formed by Jack's line of sight to the base of the tree, the lake's surface, and the building. Since the tree and building are at the lake's edge, they're level with each other and the triangle is a right triangle. The angle at the base of the tree is 30 degrees. Using trigonometric functions, we can find the unknown lengths of the triangle. The line of sight is the hypotenuse of the triangle. The height of the building where Jack sits is the side opposite the 30 degree angle. The length across the lake, 866 ft., is the adjacent side.
Cosine = adjacent/hypotenuse
Tangent = opposite/adjacent
Cos(30)= 0.8660 = 866/hypotenuse Jack's line of sight is 1000 ft.
Tan(30)= 0.5774 = opposite/866 -> Jack's height is 500 ft.
Now, we turn our attention to the triangle formed by the tree and both of Jack's lines of sight. We can see that it's an equilateral triangle, consisting of three 60 degree angles, and therefore all sides are equal. Thus the tree is 1000 ft. high.
Bonus: Jack's height was 500 ft. The largest whole multiples of twelve to divide evenly into that is 492(41x12), placing the office on the 42nd floor (41 above the first floor). Jack's perch atop the bookcase put his eyes eight feet above the floor, rounding out 500 ft.
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