Mathematics • Year 9 • Unit 2 • Lesson 13
Hyperbolas in the Wild — Sharing, Speed, Light
Use $y = k/x$ to model pizza slices per person, travel time vs speed, brightness vs distance, gear ratios, and pressure vs volume. Pull out $k$ in each situation and predict what happens when the input doubles or halves.
1. Word problems
Each scenario gives inverse variation: the product of the two quantities stays constant ($xy = k$). Build mini-tables, identify $k$, and reason about what doubling or halving does. 3 marks each
1.1 — Pizza sharing. One large pizza is split equally between $n$ people. Each person gets $s = 24/n$ slices.
(a) Build a table of $s$ for $n = 1, 2, 3, 4, 6, 8, 12$.
(b) State the value of $k$ and explain what it represents about the pizza.
(c) Describe what happens to slices per person as $n$ doubles from $2$ to $4$, then from $4$ to $8$.
1.2 — Travel time vs speed. A bus driver covers a fixed $120$ km route. Travel time $t$ (hours) and average speed $v$ (km/h) are related by $t = 120/v$.
(a) Build a table of $t$ for $v = 40, 60, 80, 120$ km/h.
(b) State $k$ and what it represents physically.
(c) What two asymptotes does the graph of $t$ vs $v$ have? What does each asymptote mean about the bus journey?
1.3 — Brightness and distance. The brightness $B$ (lumens) received from a torch at distance $d$ (m) is modelled (simplified, not strictly the inverse-square law) by $B = 48/d$.
(a) Build a table of $B$ at $d = 1, 2, 3, 4, 6, 8$.
(b) Halving the distance from $4$ m to $2$ m does what to the brightness?
(c) Which quadrant of the graph is physically meaningful here, and why? (Hint: can $d$ ever be negative?)
1.4 — Gear ratios on a bike. On a fixed road speed, the rear cog teeth $T$ and pedal cadence $c$ (revs per minute) satisfy $T \cdot c = 720$ (so $T = 720/c$).
(a) Find $T$ when the rider pedals at $c = 60$, $80$, $90$, $120$ rpm.
(b) Identify the value of $k$ and what it means.
(c) If you increase your cadence, do you need a bigger or smaller rear cog? Justify using the inverse-variation pattern.
1.5 — Pressure and volume (Boyle's law, simplified). For a sealed bicycle pump at constant temperature, the pressure $P$ (kPa) and volume $V$ (mL) of the air inside satisfy $P \cdot V = 200$, so $P = 200/V$.
(a) Compute $P$ at $V = 10, 20, 40, 50$ mL.
(b) When you compress the air from $V = 40$ to $V = 20$ mL, how does the pressure change?
(c) The graph of $P$ vs $V$ is a hyperbola. State $k$ and the asymptotes.
2. Explain your thinking
Use full sentences, no dot points. 4 marks
2.1 A classmate is sketching $y = 6/x$ and draws ONE smooth curve passing through the origin. In your own words, explain (i) why their sketch is wrong, (ii) what shape the graph of $y = 6/x$ actually has, (iii) why the curve cannot touch the $x$-axis or the $y$-axis, and (iv) what value $y$ takes at $x = 0$ and why.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pizza sharing
(a) $n = 1, 2, 3, 4, 6, 8, 12$ gives $s = 24, 12, 8, 6, 4, 3, 2$ slices.
(b) $k = 24$. This represents the total number of slices in the pizza.
(c) From $n = 2$ (12 slices each) to $n = 4$ (6 each), $s$ halves. From $n = 4$ (6 each) to $n = 8$ (3 each), $s$ halves again. Doubling the number of people always halves the slices per person.
1.2 — Travel time vs speed
(a) $v = 40, 60, 80, 120$: $t = 3, 2, 1.5, 1$ hours.
(b) $k = 120$ = total distance (km).
(c) Asymptotes: $v = 0$ (vertical) and $t = 0$ (horizontal). $v = 0$ means the bus is stopped — no fixed travel time (mathematically time blows up). $t = 0$ means no time has passed — you can approach it by driving impossibly fast, but you can never reach it on a finite route.
1.3 — Brightness and distance
(a) $d = 1, 2, 3, 4, 6, 8$ gives $B = 48, 24, 16, 12, 8, 6$.
(b) From $d = 4$ ($B = 12$) to $d = 2$ ($B = 24$): brightness doubles.
(c) Only Q1 ($d > 0$, $B > 0$) is physically meaningful, because distance can't be negative and the torch can't be "negatively bright". The Q3 branch of the hyperbola exists mathematically but has no meaning here.
1.4 — Gear ratios
(a) $c = 60: T = 12$. $c = 80: T = 9$. $c = 90: T = 8$. $c = 120: T = 6$.
(b) $k = 720$. Physically it's the product of cog teeth and cadence at this road speed — a fixed "drive constant" for the bike.
(c) Increasing cadence means a smaller rear cog (fewer teeth). Inverse variation: $T$ goes down as $c$ goes up, because $T \cdot c$ has to stay $= 720$.
1.5 — Pressure and volume
(a) $V = 10, 20, 40, 50$: $P = 20, 10, 5, 4$ kPa.
(b) From $V = 40$ ($P = 5$) to $V = 20$ ($P = 10$): pressure doubles.
(c) $k = 200$. Asymptotes: $V = 0$ (vertical) and $P = 0$ (horizontal). Both axes are off-limits to the hyperbola.
2.1 — Explain your thinking (sample response)
My classmate is wrong because $y = 6/x$ is not a single smooth curve through the origin. The graph has TWO separate branches — one in quadrant 1 (for positive $x$) and one in quadrant 3 (for negative $x$) — that never join up. The curve cannot touch the $x$-axis because that would mean $y = 0$, but $6/x = 0$ has no solution (six divided by anything is never zero). It cannot touch the $y$-axis because that would mean $x = 0$, but $6/0$ is undefined — division by zero is not allowed. So at $x = 0$, $y$ is undefined, and the graph has a gap there. The axes act as asymptotes — lines the branches approach but never cross.
Marking: 1 mark for "two separate branches"; 1 mark for explaining why $y \neq 0$ (no $x$-intercept); 1 mark for explaining why $x = 0$ is undefined; 1 mark for using the word "asymptote" correctly.