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.