SAT Question of the Week #4 Solutions:

by Oren Lahav

Hey everybody!

As I’m sure you’ve all noticed, we had a great math question out this week. Here are the solutions! But first, let’s congratulate manas biswas, who solved the question and posted it in our discussion. Hopefully next time we’ll have a more intense competition and more people will post their answers.

Here’s the question:

The side of this cube is given by the formula (1/(s/4)) + (3/2) = 2 + (1/2).

If B, C and E are all midpoints of their respective edges of the cube, find the size of the largest of these angles:

XAY

XBY

XCY

XDY

XEY

And the solution:

The equation gives (4/s) = 1 by simply moving the 3/2 over. Thus, s=4.

Just looking at the picture should give you a fairly clear idea of which angle is the largest. We’re basically looking for the shortest distance from Y, and clearly B is shorter than any of the rest- you can check by calculating them:

BY is 2 units, since B is the midpoint. AY is 4, YC and YE can be found using the Pythagorean Theorem- they both equal the squared root of 20, which is about 4.47. YD is the squared root of 32, which is 5.65.

So XY=4, and BY=2. Then by SOHCAHTOA we have tan(XBY)=4/2, and the angle is the inverse tangent of 2, which is approximately 64 degrees.

By the way, angle XAY=45, angle XDY=35, and angles XCY=XEY=42 degrees, approximately. So clearly XBY is largest.

Great job if you’ve got the answer right, and post below any concerns or mistakes.

Wait up for a cool new question next week!

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