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.