Represent the given equations C1 and C2 with the help of a diagram.

Q: How do you find the radius of a circle from $x^2+y^2=r^2$?

Compare the given equation to the standard form $x^2+y^2=r^2$; the radius is the square root of the constant on the right-hand side. Here 64 gives radius 8 and 4 gives radius 2.

Q: Why are the two circles concentric?

Both equations are of the form $x^2+y^2=\text{constant}$ with no shift in x or y, meaning both circles are centred at the origin $(0,0)$; only their radii differ.

Application of Integrals

Graphing Concentric Circles from their Equations

Official Answer

The diagram consists of two circles, both centred at the origin O, drawn on the same pair of coordinate axes.

Circle C1: $x^2 + y^2 = 64$

Centre $(0, 0)$ , radius = $\sqrt{64} = 8$ units
Passes through $(8,0)$ , $(-8,0)$ , $(0,8)$ , $(0,-8)$
Represents the outer boundary of the roundabout

Circle C2: $x^2 + y^2 = 4$

Centre $(0, 0)$ , radius = $\sqrt{4} = 2$ units
Passes through $(2,0)$ , $(-2,0)$ , $(0,2)$ , $(0,-2)$
Represents the boundary of the circular pond

Relative position: Since $2 < 8$ , C2 lies entirely inside C1 — the two circles are concentric, with the pond (C2) at the centre of the roundabout (C1), and the annular region between them is the drivable road area.

concentric circlesradiusoriginx²+y²=r²roundaboutcircular ponddiagram

Marking Scheme

11 mark: correctly drawn concentric circles centred at the origin, with radius 8 for C1 and radius 2 for C2, clearly labelled.

Hint

Both equations are of the form $x^2+y^2=r^2$ ; identify the radius as $\sqrt{\text{constant}}$ and sketch both circles centred at the origin.

Quick Oral Answer

Both equations represent circles centred at the origin — C1 has radius 8 and C2 has radius 2 — so they are concentric circles with C2 lying entirely inside C1.

Analysis & Explanation

This sub-question checks whether the equations of two circles can be correctly translated into a visual representation before integration is attempted.

Concept: The general equation $x^2+y^2=r^2$ always represents a circle centred at the origin with radius r; since both C1 and C2 have this same standard form (no shift in x or y), they must be concentric. Recognising the geometry before any calculus is essential because the rest of the case study (parts ii and iii) depends on correctly identifying the shape and its radius.

Exam trap: A common error is sketching two intersecting or offset circles instead of concentric ones, or drawing the two circles at comparable sizes; the diagram must clearly show C2 (radius 2) tucked well inside C1 (radius 8).

Real-world application: This concentric-circle model mirrors real traffic roundabout design, where a smaller non-traversable central island (sometimes landscaped, sometimes with a fountain, as here) sits at the centre of a larger circulating carriageway — used worldwide to slow traffic safely without signals.

Common Mistakes

1Drawing the two circles with different centres instead of concentric circles both centred at the origin.
2Getting the radius wrong by forgetting the square root — using 64 and 4 as the radii instead of 8 and 2.
3Not labelling which circle is C1 and which is C2, leaving the diagram ambiguous.

Interesting Facts

Real traffic-engineering roundabout design uses exactly this kind of concentric-circle geometry — the central island is deliberately made much smaller than the circulating carriageway, just as C2 is far smaller than C1 here.

The equation $x^2+y^2=r^2$ comes from coordinate geometry methods pioneered by Descartes in the 17th century, long before the modern traffic roundabout (patented in the UK in 1966).

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Frequently Asked Questions

How do you find the radius of a circle from $x^2+y^2=r^2$ ?

Compare the given equation to the standard form $x^2+y^2=r^2$ ; the radius is the square root of the constant on the right-hand side. Here 64 gives radius 8 and 4 gives radius 2.

Why are the two circles concentric?

Both equations are of the form $x^2+y^2=\text{constant}$ with no shift in x or y, meaning both circles are centred at the origin $(0,0)$ ; only their radii differ.