Let's face it. If you're not a math master, there will be at least one question on the SAT Math that will stump you. What I mean by stumping you, I mean that you will see a question that you have no idea on how to solve it.
Since the SAT is notorious for testing students on math concepts that students are likely to forget about, chances are that you will see a geometry question you aren't too sure on how to solve, or second guess yourself on a logic question.
Not that this will happen very often, but you may unfortunately end up with an "impossible to solve" SAT Math question.
Check out this bad boy:
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| Source: The Princeton Review |
I'll be honest; I don't know how to solve the "area of the shaded region". However, I do know how to find the answer.
You may be tempted to recall or look up some geometry formulas that solve for areas. And you will most likely sigh after realizing your best friend Circle Formula can't find the answer for you.
Embrace yourself, because I am about to show you the LAMEST way to find the answer.
Notice that the shaded region is inside a square. Each side of the square has a length of 20. So the area of the square is 400. (Area of square = side x side)
As you can see, the shaded region inside the square looks like it's taking up about half of the square's space. Thus we can assume that the answer has to be around 200.
(A) 20π ≈ 62.832
(B) 40(π-2) ≈ 45.664
(C) 200(π-2) ≈ 228.319
(D) 100π ≈ 314.159
(E) 400π ≈ 1256.637
The answer is (C) 200(π-2). I know that 228.319 isn't half of 400, but that is the closest value to 200. You can obviously tell that (A) and (B) are way too small, (D) is too big, and (E) is way too big.
Now you might start protesting "What if the figure isn't drawn to scale??!"
You are bringing up an absolutely important point. My method of estimating the answer wouldn't work if the figure wasn't drawn to scale. But I know for a FACT that the figure was given was drawn to scale.
Here's why:
Check out the circled number 3 in the Notes Box. It states that all figures are drawn to scale, unless stated otherwise. The question we just solved didn't have a disclaimer. Therefore, we can estimate the area.
I know that you are thinking that doing math this way isn't correct, and you are right. Even a middle school student can solve this question because most of the work is done by estimating. But whether you like it or not, how you solve the question doesn't matter. If you can guess on EVERYTHING and score a 2400, then you're golden. The point of the SAT is to score as high as possible by any means necessary (if legal). And in this case, estimating the area is how to get that extra point.
The solution does not require calculus, just basic geometry. Divide or cut up the unshaded area into 8 congruent figures (each figure looks like a right triangle with the hypotenuse curved inwards). The area of one of these figures is r^2 - π *r^2/4, where r is the radius of a circle. There are 8 of them, 8[r^2 - π *r^2/4]. Then to get the shaded area subtract this from (2r)^2, the area of the entire square.
ReplyDeleteWhence we get, after simplifying, 2*π*r^2 - 4*r^2. When r = 10 we have 200π - 400 or 200 ( π - 2).