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Test solutions Grade 10 : Area, Surface Area, and Volume — Zyro the alien explorer of Planète Maths

Test solutions with the detailed point scale. Add up your points and spot what to review.

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

1 A shed wall / 3 pts

  1. Rectangle: \(10 \cdot 6 = 60\) (0.5 pt). Triangle: \(\dfrac{1}{2} \cdot 10 \cdot 4 = 20\) (1 pt). Total 80 m² (0.5 pt).
  2. \(80 \cdot 3.50 = 280\), so the cost is 280 dollars (1 pt).

2 A fish tank / 3 pts

  1. \(V = 60 \cdot 30 \cdot 40 = 72{,}000\) cm³ = 72 L (1 pt).
  2. \(\dfrac{3}{4} \cdot 72 = 54\) L (1 pt).
  3. Total \(2(1800 + 2400 + 1200) = 10{,}800\), minus the top 1,800, gives 9,000 cm² (1 pt).

3 Cylinder and cone / 4 pts

  1. \(V = \pi \cdot 16 \cdot 9 = 144\pi \approx 452.39\) cm³ (1 pt).
  2. \(V = \dfrac{1}{3} \cdot 144\pi = 48\pi \approx 150.80\) cm³ (1 pt).
  3. \(\text{SA} = 2\pi \cdot 16 + 2\pi \cdot 4 \cdot 9 = 32\pi + 72\pi = 104\pi \approx 326.73\) cm² (1 pt).
  4. \(\dfrac{144\pi}{48\pi} = 3\) times (1 pt).

4 A ball / 3 pts

  1. The radius is 12 cm (0.5 pt). \(\text{SA} = 4\pi \cdot 144 = 576\pi\) (0.75 pt). \(V = \dfrac{4}{3}\pi \cdot 1728 = 2304\pi\) (0.75 pt).
  2. \(2304\pi \approx 7238.23\) cm³ \(\approx 7.24\) L (1 pt).

5 Slicing a pyramid / 4 pts

  1. \(V = \dfrac{1}{3} \cdot 256 \cdot 12 = 1024\) cm³ (1 pt).
  2. The plane is 9 cm below the apex, so \(k = \dfrac{9}{12} = 0.75\) (1 pt). The side is \(16 \cdot 0.75 = 12\) cm and the area is 144 cm² (1 pt).
  3. \(V = \dfrac{1}{3} \cdot 144 \cdot 9 = 432\) cm³ (1 pt), which equals \(1024 \cdot 0.75^3\).

6 Steel ball and Cavalieri / 3 pts

  1. \(V = \dfrac{4}{3}\pi \cdot 27 = 36\pi \approx 113.10\) cm³ (1 pt).
  2. \(m \approx 113.10 \cdot 7.85 \approx 887.8\) g (1 pt).
  3. Both stacks lie between the same two planes, and every plane parallel to them cuts each stack in a rectangle of the same area, so Cavalieri’s principle says the volumes are equal (1 pt).
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