Mathematics • Year 9 • Unit 2 • Lesson 17
Real-World Non-Linear Scenarios
Five real applications: a diver from a platform, friends sharing rent, a savings account growing, a Ferris wheel arc, and a tossed apple. Identify the family, extract real features (with units), and explain what they mean.
1. Word problems
For each: identify the family, do the calculation, then state the real-world meaning of each feature (with units, rejecting impossibles). 3 marks each
1.1 — Diver from a $10$ m platform. A diver's height (m) above water $t$ seconds after launch is $h = -5t^2 + 5t + 10$.
(a) State the diver's launch height (substitute $t = 0$).
(b) Find $t$ when $h = 0$ (entry to water) by factoring $-5(t^2 - t - 2) = 0$ and rejecting the negative root.
(c) Use the axis of symmetry (midpoint of the two roots) to find the time of maximum height, then compute that height.
1.2 — Sharing rent. Six friends share a $\$300$ accommodation cost. Each friend's cost is $C = \dfrac{300}{n}$ where $n$ is the number of friends.
(a) Compute $C$ for $n = 3, 6, 10$ (with the dollar sign).
(b) Name the family.
(c) What does the horizontal asymptote $C = 0$ mean in this context?
1.3 — Savings account. An account starts at $\$500$ and grows by factor $1.1$ each year: $A = 500 \cdot 1.1^t$.
(a) Compute $A$ after $1$ year and $2$ years (round to the nearest dollar).
(b) Identify the family and the meaning of "$A(0) = 500$" in the real world.
(c) Explain why "grows by factor $1.1$" is a $10\%$ annual increase.
1.4 — Ferris wheel. A Ferris wheel of radius $20$ m is centred at the origin (the centre of the wheel). The path of a cabin is the circle $x^2 + y^2 = 400$.
(a) State the radius and verify the equation by substituting $(20, 0)$ and $(0, -20)$.
(b) Name two points on the wheel where the cabin is at its highest, lowest, leftmost, and rightmost positions.
(c) Why is $y = f(x)$ form impossible for a circle? (One sentence.)
1.5 — Tossed apple. An apple is tossed from $h = 1$ m at $4$ m/s upward. Its height (using $g = 10$ m/s²) is $h = -5t^2 + 4t + 1$.
(a) Compute the height at $t = 0.5$ s.
(b) Find when $h = 0$ (catch or splat): factor $-5t^2 + 4t + 1 = -(5t^2 - 4t - 1) = -(5t + 1)(t - 1)$.
(c) Reject the impossible root and state the landing time with units.
2. Explain your thinking
Use full sentences. 4 marks
2.1 A friend computes the maximum height of a thrown ball as "$20$" and writes that as the final answer. In your own words, explain (i) what units are missing and why marks would be lost, (ii) why every answer in a real-world non-linear problem must include units, and (iii) give an example where the same numeric answer ($20$) could mean something completely different depending on the units (e.g. $20$ m vs $20$ km vs $20$ seconds vs $\$20$).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Diver
(a) $h(0) = -5(0) + 5(0) + 10 = 10$ m — diver launches from $10$ m platform. ✓
(b) $-5t^2 + 5t + 10 = 0 \Rightarrow t^2 - t - 2 = 0 \Rightarrow (t - 2)(t + 1) = 0$. So $t = 2$ or $t = -1$. Reject $t = -1$ (negative time impossible). Diver enters water at $t = 2$ s.
(c) Axis of symmetry: midpoint of $-1$ and $2$ is $t = 0.5$ s. Max height: $h(0.5) = -5(0.25) + 5(0.5) + 10 = -1.25 + 2.5 + 10 = 11.25$ m.
1.2 — Sharing rent
(a) $C(3) = \dfrac{300}{3} = \$100$. $C(6) = \dfrac{300}{6} = \$50$. $C(10) = \dfrac{300}{10} = \$30$.
(b) Family: hyperbola (inverse proportion, $Cn = 300$).
(c) The horizontal asymptote $C = 0$ means: as the number of friends sharing grows very large, the per-person cost approaches (but never reaches) zero dollars. You can never share a $\$300$ bill so wide that any one person pays exactly $\$0$ — but with $300{,}000$ friends each would pay $0.1$ cent.
1.3 — Savings account
(a) $A(1) = 500 \times 1.1 = \$550$. $A(2) = 500 \times 1.21 = \$605$.
(b) Family: exponential growth. "$A(0) = 500$" is the initial deposit — the balance at the moment the account was opened, before any interest.
(c) "Grows by factor $1.1$" means the new balance is $1.1 \times $ the old balance. Since $1.1 = 1 + 0.1$, the new balance equals the old balance PLUS $10\%$ of the old balance. That's a $10\%$ annual increase, compounded.
1.4 — Ferris wheel
(a) Radius $= \sqrt{400} = 20$ m. Check $(20, 0)$: $20^2 + 0^2 = 400$ ✓. Check $(0, -20)$: $0^2 + (-20)^2 = 400$ ✓.
(b) Highest: $(0, 20)$. Lowest: $(0, -20)$. Rightmost: $(20, 0)$. Leftmost: $(-20, 0)$.
(c) A circle fails the vertical line test — for most $x$-values there are TWO $y$-values (upper and lower halves), so it can't be written as a single $y = f(x)$ rule.
1.5 — Tossed apple
(a) $h(0.5) = -5(0.25) + 4(0.5) + 1 = -1.25 + 2 + 1 = 1.75$ m.
(b) $-(5t + 1)(t - 1) = 0 \Rightarrow 5t + 1 = 0$ or $t - 1 = 0$, so $t = -\tfrac{1}{5}$ or $t = 1$.
(c) Reject $t = -\tfrac{1}{5}$ (time can't be negative). Landing time $= 1$ second.
2.1 — Explain your thinking (sample response)
The friend's answer is missing the unit "metres". In a height problem, the units could realistically be metres, centimetres or feet — without one of those stated, the answer is ambiguous and marks will be lost. Every real-world answer needs units because the same number can mean very different things in different contexts: $20$ m is the height of a small apartment building; $20$ km is a long bike ride; $20$ seconds is the time it takes to brush your teeth; $\$20$ is the cost of a takeaway meal. The number alone tells you almost nothing about the real-world situation — the units are doing most of the meaningful work. The correct answer is "$20$ metres at $t = 2$ seconds" — number AND unit AND a statement of when.
Marking: 1 mark for naming the missing unit ("metres"); 1 mark for "every real-world answer needs units"; 1 mark for a clear example of the same number meaning different things; 1 mark for clear, full-sentence writing.