Christmas Tree Math Problem

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So today I have a special holiday question for you guys. Christmas is coming up, and, as some of you may know, Rockefeller Center in New York puts up one of the largest Christmas trees in the country. The tree is decorated with a Swarovski crystal star and has become a symbol for the holiday season in Times Square.

To get in the holiday spirit, I came up with this fun SAT trigonometry problem about the Rockefeller Center Christmas tree.

The Rockefeller Center Christmas tree has a star 9.5 feet tall. Around noon, the shadow of the tree reaches only 33.3 feet to the nearby ice skating rink. An observer standing on the edge of the shadow looks up at the tree and measures the angle of inclination as 68.5°. About how tall, in feet, is the Christmas tree (round to nearest foot)?

(A) 75
(B) 77
(C) 82
(D) 85

Figure for SAT Trigonometry Problem

We’ve been given some information about the tree, and we’re asked to calculate its height. From the figure, we can see that we’re dealing with a right triangle problem. And, because we have a single angle and the length of one side, we must be dealing with an SAT trigonometry question.

Now, SAT trigonometry questions will always be very simple. The test makers only expect you to know a handful of identities, so you can easily solve these questions by analyzing the information given and the information needed. Here, we have an angle and the length of its adjacent side, and we need to find the length of the opposite side.

SOH: Sin(x) = Opposite/Hypotenuse
CAH: Cos(x) = Adjacent/Hypotenuse
TOA: Tan(x) = Opoosite/Adjacent

Thinking back to SOH-CAH-TOA, we see that taking the tangent of the given angle will give us the ratio of the opposite and adjacent sides. Now, we plug in the given angle and the length of the adjacent side, then we simplify the equation to solve for the length of the opposite side, which is equal to the height of the Christmas tree.

tan(x) = opposite / adjacent (TOA)
tan(68.5) = opposite / 33.3
33.3 tan(68.5) = opposite
Opposite = 84.5 feet

Remember that the star on top is 9.5 feet, so be sure to subtract 9.5 from the length of the opposite side to calculate the height of the tree itself. Now, let’s review our answer choices. Seventy-five, of course, is choice (A).

And if you notice, choice (D) is 85, which is the length of the tree and the star, rounded up. So even with a quick and simple question like this, it’s important to read the question thoroughly. Always make sure you actually answer the question and not just solve for a value.

Happy Holidays, and thanks for reading!


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