Mathematics • Year 9 • Unit 2 • Lesson 12

Circles in Real Life — Wi-Fi, Pizza, Sprinklers

Use $x^2 + y^2 = r^2$ to model a wifi router's coverage, a pizza, a garden sprinkler, a Ferris wheel, and a phone GPS lock. Decide whether key points are inside, on, or outside.

Apply · Real-World Maths

1. Word problems

Treat the origin $(0, 0)$ as the centre of each circular region (router, pizza, etc). Use $x^2 + y^2 = r^2$ and the on/inside/outside test ($< r^2$ inside, $= r^2$ on, $> r^2$ outside). 3 marks each

1.1 — Wi-Fi router. A Wi-Fi router sits at the origin of a coordinate grid (units in metres) and reaches every point inside a $10$-metre radius.

(a) Write the equation of the boundary circle.
(b) Is a laptop at $(6, 8)$ inside, on, or outside the coverage area? Show your working.
(c) Is a phone at $(5, 9)$ getting signal? Justify.

Stuck? Boundary equation has $r^2 = 100$. Compare $6^2 + 8^2 = 100$ and $5^2 + 9^2 = 106$ with $100$.

1.2 — Pizza. A circular pizza of radius $20$ cm is centred at the origin of a coordinate grid (in centimetres) printed on the cutting board.

(a) Write the equation of the edge of the pizza.
(b) A piece of pepperoni sits at $(12, 16)$. Is it on the pizza or has it fallen off the edge?
(c) A second piece is at $(15, 14)$. Is it on the pizza? Show working.

Stuck? $r^2 = 400$. Compute each point's $x^2 + y^2$ and compare with $400$.

1.3 — Garden sprinkler. A garden sprinkler at the origin throws water in a circle of radius $3$ m. A tomato plant is at $(2, 2)$.

(a) Write the equation of the watered boundary.
(b) Does the tomato plant get watered? Justify with the inside/on/outside test.
(c) The gardener wants to also water a basil plant at $(3, 3)$. What is the smallest whole-metre radius the sprinkler would need to set to JUST reach it (round up)? Write the new equation.

Stuck on (c)? For the basil to be on or inside, we need $r^2 \ge 18$. Smallest whole $r$ with $r^2 \ge 18$ is $r = 5$ (since $4^2 = 16 < 18$).

1.4 — Ferris wheel. A Ferris wheel of radius $15$ m turns around a centre at the origin (height in metres). Treat each carriage as a point on the rim.

(a) Write the equation of the rim.
(b) At one moment, a carriage is at the top: what are its coordinates? List the four equivalent "axis" positions (top, bottom, far right, far left).
(c) Verify that the point $(9, 12)$ lies on the rim by substitution.

Stuck on (c)? Compute $9^2 + 12^2 = 81 + 144 = 225 = 15^2$. So yes, on the rim.

1.5 — Phone GPS uncertainty. A phone's GPS lock places you somewhere inside a circle of radius $5$ m around the reported point. Place the reported point at the origin (units in metres).

(a) Write the equation describing the "edge of the uncertainty disc".
(b) A friend says you might actually be at $(4, 4)$. Is this point inside the uncertainty disc? Justify.
(c) Could you actually be at $(0, 5)$? At $(6, 0)$? Decide for each and explain.

Stuck? $r^2 = 25$. Compare $4^2 + 4^2 = 32$, $0^2 + 5^2 = 25$, $6^2 + 0^2 = 36$ with $25$.

2. Explain your thinking

Use full sentences, no dot points. 4 marks

2.1 A classmate sees the equation $x^2 + y^2 = 36$ and says "the radius is $36$." In your own words, explain (i) why their answer is wrong, (ii) what the right side of a circle equation actually represents, (iii) the correct radius, and (iv) how you would convince them using a quick point-substitution that $r = 6$ (not $36$).

Stuck? Revisit lesson § "Spot the Trap" — the first wrong/right pair is exactly this misconception. Then substitute $(6, 0)$ into the equation to test $r = 6$.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Wi-Fi router

(a) $\mathbf{x^2 + y^2 = 100}$ (since $r = 10$, $r^2 = 100$).
(b) $6^2 + 8^2 = 36 + 64 = 100 = r^2$, so the laptop is exactly ON the boundary (just at the edge of coverage).
(c) $5^2 + 9^2 = 25 + 81 = 106 > 100$, so the phone is OUTSIDE — no signal.

1.2 — Pizza

(a) $\mathbf{x^2 + y^2 = 400}$ (cm²).
(b) $12^2 + 16^2 = 144 + 256 = 400 = r^2$, so the pepperoni is exactly ON the edge of the pizza.
(c) $15^2 + 14^2 = 225 + 196 = 421 > 400$, so this piece has fallen OFF.

1.3 — Garden sprinkler

(a) $\mathbf{x^2 + y^2 = 9}$.
(b) Tomato at $(2, 2)$: $4 + 4 = 8 < 9$, so INSIDE — gets watered.
(c) Basil at $(3, 3)$: $9 + 9 = 18$. Need $r^2 \ge 18$. $r = 4$ gives $r^2 = 16$ (not enough); $r = 5$ gives $r^2 = 25 \ge 18$. Smallest whole-metre radius is $\mathbf{r = 5}$ m, new equation $\mathbf{x^2 + y^2 = 25}$.

1.4 — Ferris wheel

(a) $\mathbf{x^2 + y^2 = 225}$ (since $r = 15$, $r^2 = 225$).
(b) Top $(0, 15)$, bottom $(0, -15)$, far right $(15, 0)$, far left $(-15, 0)$.
(c) $9^2 + 12^2 = 81 + 144 = 225 = r^2$. Yes, $(9, 12)$ lies exactly on the rim.

1.5 — Phone GPS uncertainty

(a) $\mathbf{x^2 + y^2 = 25}$.
(b) $4^2 + 4^2 = 32 > 25$: OUTSIDE. So $(4, 4)$ is NOT inside the uncertainty disc.
(c) $(0, 5)$: $0 + 25 = 25 = r^2$ — ON the edge (just possible). $(6, 0)$: $36 + 0 = 36 > 25$ — OUTSIDE (impossible according to the GPS).

2.1 — Explain your thinking (sample response)

My classmate is wrong because the right side of a circle equation is $r^2$, not $r$ itself. In $x^2 + y^2 = 36$, the $36$ is the radius SQUARED, so the actual radius is $r = \sqrt{36} = 6$. They need to take the square root of the right side every time. To convince them, I would substitute a point that should sit on the rim: at $(6, 0)$, the equation gives $6^2 + 0^2 = 36$, which equals the right side, so $(6, 0)$ is on the circle. That confirms the rim is $6$ units from the centre — in other words, $r = 6$. If the radius were really $36$, the point $(6, 0)$ would be deep inside, not on the rim.

Marking: 1 mark for naming "right side is $r^2$"; 1 mark for $r = 6$ via square root; 1 mark for the substitution check; 1 mark for clear, full-sentence writing.