A wooden duster has dimensions 5 cm × 5 cm × 7 cm and is gently lowered into a cylindrical vessel of water with radius 5 cm and height 10 cm. After inserting the duster, the water level rises by 2.14 cm. What percentage of the volume of the duster remains above the water surface?

Introduction / Context:
This question involves volume displacement and percentage comparison. By placing a solid object (the duster) into water, some part of the object is submerged and displaces water, which raises the water level. Using the concept that the volume of water displaced equals the submerged volume of the object, we can determine what fraction of the object is under water and therefore what fraction remains above the surface.

Given Data / Assumptions:

• Dimensions of the duster: 5 cm × 5 cm × 7 cm.
• Shape of the container: a cylinder with radius 5 cm and height 10 cm.
• Rise in water level after inserting the duster = 2.14 cm.
• The duster floats partially submerged; it is not completely sunk to the bottom in the sense of being fully under water.
• We neglect any water spillage or absorption and assume perfect displacement.

Concept / Approach:
The principle of displacement tells us that the volume of water displaced equals the volume of the object that is submerged. The rise in water level within a cylindrical vessel directly gives the displaced volume as base area * height rise. Once we know the submerged volume, we compare it with the total volume of the duster to find the submerged fraction. The remaining fraction, multiplied by 100, gives the percentage of the duster that remains above the water surface.

Step-by-Step Solution:
Step 1: Calculate total volume of the duster. Volume of duster = 5 * 5 * 7 = 175 cubic centimetres. Step 2: Calculate volume of water displaced. Base area of cylinder = pi * r^2 = pi * 5^2 = 25 * pi square centimetres. Height rise = 2.14 cm. Displaced volume = base area * height rise = 25 * pi * 2.14 cubic centimetres. Step 3: Approximate pi as 3.14. Displaced volume ≈ 25 * 3.14 * 2.14 ≈ 168 cubic centimetres (approximately). Step 4: This displaced volume equals the submerged volume of the duster, so submerged volume ≈ 168 cubic centimetres. Step 5: Fraction submerged = 168 / 175 ≈ 0.96, that is 96%. Step 6: Therefore, fraction above the water = 1 - 0.96 = 0.04, which is 4%.

Verification / Alternative check:
We can verify the fraction using exact symbolic reasoning. Let submerged fraction be x. Then x * (volume of duster) = volume of displaced water. So x * 175 = 25 * pi * 2.14. Solve for x: x = (25 * pi * 2.14) / 175. Note that 25 / 175 = 1 / 7, so x = (pi * 2.14) / 7. Taking pi ≈ 3.14, x ≈ (3.14 * 2.14) / 7 ≈ 6.7196 / 7 ≈ 0.96. This again shows that 96% of the duster volume is submerged and 4% remains above water.

Why Other Options Are Wrong:
96% would represent the proportion submerged, not the part above the water. The question explicitly asks for the portion above the layer, that is the complement.
97% and 3% do not correspond to the accurate calculations and would require a different displaced volume than the one derived from the given rise of 2.14 cm.

Common Pitfalls:
A common mistake is to interpret the percentage as the submerged part instead of the part above the water. Another error is in calculating the displaced volume, either by forgetting to square the radius in the area formula or by misusing the value of pi. Some students also confuse the absolute volume difference (175 - 168) with the percent directly, without dividing by the total volume first. Always work systematically: compute volumes, find the fraction, then convert to a percentage and interpret correctly in context.

Final Answer:
Approximately 4% of the duster's volume remains above the water surface.