NOT A REAL PLANE ON FIRE. Source.

Appropriately for May Day, this question will cause you to send distress signals. Prize this week: you get to strut around all weekend telling people how much smarter you are than they, and they have to humbly agree. You may force up to three strangers to “kiss the ring.”

Note: Figure not drawn to scale.

In the figure above, P is the center of a circle, and Q and R lie on the circle. If the area of the sector PQR is 25π³ and the length of arc QR is 5π², what is the measure of ∠RPQ?

Put your answers in the comments. I’ll post the solution Monday. Good luck!

UPDATE: props and ring kisses to Elias, who answered on Facebook, and to JD, who did his thing right here on the site (and spelled out the solution quite elegantly). Side note: I’d love to be able to integrate those two commenting systems — is that possible?

Solution below the cut:

Solution

This difficult circle question its, at its core, a ratio question. Remember that, with circles, part/whole = part/whole, or:

(For a more rigorous explanation and practice with this, see this post.)

We can set set up ratios using this framework for both the area of the sector and the length of the arc:

At this point, you could just use Wolfram Alpha, but surely you’d rather do it the long way. I know your pulse quickened a bit when you noticed that both equations share the same left side, and can therefore be combined:

And neatly solved for r:

Sweet. If r = 10π, we can now solve for x using one of our original ratios: