Topic Review — Problem Solving and Modelling — Solutions

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1. Mathematical modelling

   1. Electrician: C = 90 + 65h. For h=2.5: C = 90+162.50 = $252.50
   2. Tank draining: V = 6000 − 150t. Empty when V=0: t = 6000/150 = 40 minutes
   3. Assumption and limitation: Assumption: drainage rate is constant throughout. Limitation: in reality, flow rate often decreases as water pressure drops when the tank empties, making a linear model an approximation.
   4. Equal cost at MB: 45 + 0.02M = 60 → 0.02M = 15 → M = 750 MB. Below 750 MB, Plan A is cheaper; above, Plan B is better.

2. Multi-topic fluency

   1. √48+2√12−√75: 4√3+4√3−5√3 = 3√3
   2. 5(x−2)=3x+4: 5x−10=3x+4 → 2x=14 → x=7
   3. Triangle with legs 5 and 12: Hypotenuse=√(25+144)=√169=13 cm. Angle opposite 12: θ=tan&sup-1;(12/5)≈tan&sup-1;(2.4)≈67.4°
   4. Cylinder volume: r=5 cm. V=π×25×8=200π cm³ ≈ 628.32 cm³
   5. Laptop sale price: $2200 × 0.85 = $1 870

3. Rectangular room dimensions

   1. Equation: (x+4)(x−1) = 36 → x²+3x−4=36
   2. Quadratic: x²+3x−40=0 ✓
   3. Factorise: (x+8)(x−5)=0 → x=5 (taking positive root)
   4. Dimensions and perimeter: Length=5+4=9 m, Width=5−1=4 m. Perimeter=2(9+4)=26 m
   5. Carpet cost: Area=9×4=36 m². Cost=36×$45=$1 620

4. Candle profit model

   1. Meaning of −120: −$120 represents the fixed costs (e.g. materials, equipment) that must be covered before any profit is made.
   2. Break-even: 8x−120=0 → x=15. 15 candles must be sold.
   3. Profit at 30 candles: P=8(30)−120=240−120=$120
   4. Model appropriateness: Linear is appropriate if each candle sells for the same price ($8 profit per candle). Assumption: all candles sell at the same price with no unsold stock. Limitation: bulk discounts, variable costs, or unsold items would make the model less accurate at larger quantities.

5. Wheelchair ramp

   1. Minimum horizontal length: tan(5°)=0.75/d → d=0.75/tan(5°)=0.75/0.08749≈8.57 m
   2. Ramp surface length: hyp=√(8.57²+0.75²)=√(73.44+0.5625)=√74.00≈8.60 m
   3. Ramp area: Area=8.60×1.2=10.32 m²
   4. Total cost: 10.32×$85=$877.20
   5. Assumptions: (1) The ramp is perfectly straight with no curves or landings. (2) The ground is perfectly flat and level. These assumptions matter because real-world ramp installations often require level landings at top and bottom, and may need to navigate uneven ground, requiring additional material and changing the geometry significantly.