Mathematics • Year 9 • Unit 2 • Lesson 17
Applications — Mixed Challenge
Six mixed-context problems spanning projectile, inverse proportion, exponential growth/decay and circles. One classic "negative-time" mistake to catch. One open-ended design task: invent a scenario for each family.
1. Mixed applications
Show your working, include units, reject impossible answers. 3 marks each
1.1 A water rocket has height $h = -5t^2 + 25t$ (m, s). Find (a) the launch time and the landing time, (b) the time of maximum height, (c) the maximum height (with units).
1.2 A radioactive sample of $80$ g halves every $5$ years: $M = 80 \cdot \left(\tfrac{1}{2}\right)^{t/5}$ (g, years). Compute the mass at (a) $t = 5$, (b) $t = 10$, (c) $t = 20$. Name the family.
1.3 A water tank of capacity $V$ litres is filled in $T$ hours by a pump delivering at rate $R$ L/h. For a fixed $V = 600$ L, $T = \dfrac{600}{R}$. (a) Compute $T$ for $R = 20, 30, 60, 100$ L/h. (b) Why is the domain $R > 0$? (c) Name the family.
1.4 A circular fountain has equation $x^2 + y^2 = 49$ (metres). State the radius and find the coordinates of the four "compass" points (north, south, east, west of the centre, sitting on the fountain's edge).
1.5 A bridge arch is parabolic, $20$ m wide at the base, $8$ m tall at the centre. (a) If the base centre is at $x = 0$, what are the two $x$-intercepts? (b) Using $y = a(x - 0)^2 + 8$ and the intercept $(10, 0)$, find $a$. (c) Write the equation of the arch.
1.6 A bacteria culture starts at $200$ cells and doubles every hour: $N = 200 \cdot 2^t$. (a) Find $N$ at $t = 3$. (b) After how many WHOLE hours does the count first exceed $5000$? (c) Identify the family and the equation of the horizontal asymptote (if any).
2. Find the mistake
A student attempted to find the landing time of a projectile. Their working is shown. Identify the error, explain why it's wrong, and write the correct answer with units. 3 marks
Student's working for projectile $h = -5t^2 + 15t + 20$:
Set $h = 0$: $-5t^2 + 15t + 20 = 0$
Divide by $-5$: $t^2 - 3t - 4 = 0$
Factor: $(t - 4)(t + 1) = 0$
So $t = 4$ OR $t = -1$.
Student's final answer: "The projectile lands at $t = 4$ and $t = -1$."
(a) Identify the mistake.
(b) Explain in one sentence why the student's reasoning is wrong.
(c) Write the correct final answer with units. What does the rejected root represent geometrically (even though it's physically impossible)?
Stuck? Lesson § "Spot the Trap" — "Using both $x$-intercepts as landing times" is the second row.3. Open-ended challenge — invent the scenarios
This question has many valid answers. 4 marks
3.1 Invent ONE original, realistic Year 9 word problem for EACH of the four families.
For each problem:
(i) Write the scenario in $2$–$3$ sentences.
(ii) State the equation that models it.
(iii) Identify the family and one key feature (with units).
(iv) Write ONE question someone could be asked about the scenario.
The four families: parabola (projectile or bridge arch); hyperbola (inverse proportion); exponential (growth or decay); circle (circular path or boundary).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Water rocket
(a) Set $h = 0$: $-5t^2 + 25t = 0 \Rightarrow -5t(t - 5) = 0 \Rightarrow t = 0$ (launch) or $t = 5$ s (landing).
(b) Axis of symmetry: $t = \dfrac{0 + 5}{2} = 2.5$ s.
(c) Max height: $h(2.5) = -5(6.25) + 25(2.5) = -31.25 + 62.5 = 31.25$ m.
1.2 — Radioactive decay
(a) $M(5) = 80 \cdot (\tfrac{1}{2})^1 = 40$ g.
(b) $M(10) = 80 \cdot (\tfrac{1}{2})^2 = 80 \cdot \tfrac{1}{4} = 20$ g.
(c) $M(20) = 80 \cdot (\tfrac{1}{2})^4 = 80 \cdot \tfrac{1}{16} = 5$ g.
Family: exponential decay (base $\tfrac{1}{2} < 1$).
1.3 — Tank fill rate
(a) $T(20) = 30$ h. $T(30) = 20$ h. $T(60) = 10$ h. $T(100) = 6$ h.
(b) Domain $R > 0$ because a pump can't deliver at zero or negative rate — if $R = 0$ the tank never fills (division by zero), and $R < 0$ would mean the pump runs backwards (drains the tank), changing the model entirely.
(c) Family: hyperbola ($T \cdot R = 600$ is the constant-product signature).
1.4 — Fountain circle
Radius $= \sqrt{49} = 7$ m. Compass points: North $(0, 7)$, South $(0, -7)$, East $(7, 0)$, West $(-7, 0)$. (All four are obtained by setting one coordinate to zero and solving for the other: $0^2 + y^2 = 49 \Rightarrow y = \pm 7$; $x^2 + 0^2 = 49 \Rightarrow x = \pm 7$.)
1.5 — Bridge arch
(a) $x$-intercepts at $x = -10$ and $x = 10$ (base is $20$ m wide, centred at $x = 0$).
(b) Sub $(10, 0)$ into $y = ax^2 + 8$: $0 = 100a + 8 \Rightarrow a = -\dfrac{8}{100} = -0.08$.
(c) Equation of the arch: $y = -0.08x^2 + 8$ (with $y$ measured in metres). $a < 0$ confirms the arch opens downward.
1.6 — Bacteria $N = 200 \cdot 2^t$
(a) $N(3) = 200 \times 8 = 1600$ cells.
(b) $N(4) = 3200$ (still below $5000$). $N(5) = 6400$ (above $5000$). First whole hour exceeding $5000$: $t = 5$ hours.
(c) Family: exponential growth (base $2 > 1$). Horizontal asymptote: $N = 0$ (for very negative $t$, $N$ approaches but never reaches $0$).
2 — Find the mistake (negative time)
(a) The student reported BOTH solutions ($t = 4$ and $t = -1$) as landing times instead of rejecting the negative one.
(b) Time cannot be negative — $t = -1$ would mean the projectile was at ground level $1$ second BEFORE it was launched, which is physically impossible. The correct procedure is to reject negative time values when the question asks for a landing (or any physical) time.
(c) Correct answer: the projectile lands at $t = 4$ seconds. Geometrically, $t = -1$ is still a real $x$-intercept of the parabola $y = -5t^2 + 15t + 20$ — it tells you where the parabola WOULD have crossed the $t$-axis if the projectile had been moving along this path for negative time. The parabola exists for all $t$; the physical motion does not.
3 — Open-ended challenge (sample scenarios)
Parabola — kicked soccer ball. A goalkeeper kicks a ball from the ground. Its height (in m) after $t$ seconds is $h = -5t^2 + 20t$. Family: parabola, max height $= 20$ m at $t = 2$ s. Question: when does the ball land?
Hyperbola — splitting a pizza. A large pizza costs $\$24$ and is split equally between $n$ friends; each pays $C = \dfrac{24}{n}$ dollars. Family: hyperbola, asymptote $C = 0$ (more friends $\to$ smaller share). Question: what does each person pay if $8$ friends share?
Exponential — viral video. A video has $10$ views at $t = 0$ days. Each day the view count quadruples: $V = 10 \cdot 4^t$. Family: exponential growth, $V$-intercept $(0, 10)$. Question: after how many days does $V$ first exceed $1000$ views?
Circle — athletics track. A circular athletics track has radius $50$ m, centred at the origin. Equation: $x^2 + y^2 = 2500$. Family: circle, radius $50$ m. Question: if a runner starts at $(50, 0)$, what are the coordinates of the point exactly opposite on the track?
Marking: 1 mark per realistic, complete scenario with equation + family + feature + question. Award full marks for any four valid, distinct family scenarios.