The original question includes a diagram: Five sketches, each showing a family of concentric circles centred at the origin, differing in how many circles there are and how their radii are spaced.
Which of the following sketches shows the graph of sin(x2 + y2) = 12 where x2 + y2 ≤ 8π?
All five options show concentric circles centred on the origin, differing only in how many circles there are and how their radii are spaced.
- AEight circles: one small inner circle, then a wide gap, with the rest bunching more and more tightly towards the outside
- BEight circles plus a dot at the origin, spaced most widely near the outside
- CEight circles spaced roughly evenly
- DEight circles crowded together in the middle, with a large empty disc at the centre
- EFour circles plus a dot at the origin, spreading further apart towards the outside
Show the answer and worked solution
answer · A
- AEight circles: one small inner circle, then a wide gap, with the rest bunching more and more tightly towards the outside
- BEight circles plus a dot at the origin, spaced most widely near the outside
- CEight circles spaced roughly evenly
- DEight circles crowded together in the middle, with a large empty disc at the centre
- EFour circles plus a dot at the origin, spreading further apart towards the outside
The equation involves only u = x2 + y2, so the graph is a set of circles about the origin, one for each solution of sin u = 12 with 0 ≤ u ≤ 8π. Those are u = π6, 5π6, 13π6, 17π6, 25π6, 29π6, 37π6, 41π6 — eight of them, since the next, 49π6, exceeds 8π. The radii are √u, giving roughly 0.72, 1.62, 2.61, 2.98, 3.62, 3.90, 4.40, 4.63: the square root compresses the outer circles together while leaving a wide gap after the innermost one. So the right sketch has eight circles, a lone small one at the centre, and increasingly tight spacing outwards.