Give the dimension of a cuboid whose volume is halved when its surface area is doubled.
Step-by-step solution
Idea: Changing all sides by the same factor cannot work (doubling the area multiplies the volume by about 2.8). So flatten the cuboid: make it much thinner and wider. Try a square cuboid a × a × h becoming 2a × 2a × h8: its volume is 4 × 18 = 12 of the old one.
- Take a cuboid a × a × h and change it to 2a × 2a × h8. New volume = 4a2 × h8 = a2h2: halved ✓.1 mark
- Old surface area = 2a2 + 4ah. New = 2(2a)2 + 4(2a)(h8) = 8a2 + ah. For it to be doubled: 8a2 + ah = 4a2 + 8ah ⇒ 4a2 = 7ah ⇒ h = 4a7.1 mark
- Choose a = 14 cm: h = 8 cm. The cuboid is 8 cm × 14 cm × 14 cm.1 mark
- Check: old V = 8 × 14 × 14 = 1568 cm3, S = 2(112 + 196 + 112) = 840 cm2. New (1 × 28 × 28): V = 784 cm3 = 15682 ✓, S = 2(28 + 784 + 28) = 1680 cm2 = 2 × 840 ✓.1 mark
Check: 840 × 2 = 1680 ✓ and 1568 ÷ 2 = 784 ✓.
Answer to write in the exam
Cuboid a × a × h → 2a × 2a × h8: V’ = a2h2
S’ = 8a2 + ah = 2(2a2 + 4ah) ⇒ h = 4a7
a = 14 ⇒ h = 8: cuboid 8 cm × 14 cm × 14 cm
Check: V = 1568 → 784 (halved); S = 840 → 1680 (doubled)
∴ Cuboid 8 cm × 14 cm × 14 cm (becomes 1 cm × 28 cm × 28 cm)
Common mistakes that cost marks
- Scaling every side by the same factor; then volume and surface area always grow or shrink together.
- Checking only the volume or only the surface area.
How this can come in the exam
Every edge of a cuboid is doubled. Its surface area becomes
- 2 times
- 4 times
- 6 times
- 8 times
Show answer
(B) 4 times
Each face area is multiplied by 2 × 2 = 4.
Try one yourself
Show that a 2 × 3 × 6 cuboid and a 1 × 4 × 9 cuboid have the same volume. Which has the larger surface area?
Show answer
Both 36. Surface areas: 2(6 + 18 + 12) = 72 and 2(4 + 36 + 9) = 98; the flatter 1 × 4 × 9 cuboid has more.
More questions like this
- Project: Find the volume of your house making necessary approximations. Present how you solved it.
- What is the total surface area of a cuboid? You may refer to the net of the cuboid given in the figure.
- Compare the formula for the volume of a cuboid with the formula for the area of a rectangle (length × width).
- Try to work out for yourself why this model explains the formula for the volume of a cuboid, i.e.,
volume = area of base × height.
Write the formula for the surface area and the volume of a cube. - Observe how these two formulas are special cases of the formulas 2(wl + hl + hw) and whl, when w = h = l.