Volume & Surface Areas Questions and Answers

Answer: C

Recasting a solid rearranges the same material into a different shape. No matter is added or removed, so the volume is conserved:

\[ V_{\text{new}} = V_{\text{old}} \]

This is the principle behind every "melted and recast" problem - it is the equation you set up first. The surface area, by contrast, does change, and usually increases when a compact shape is drawn out into a long thin one.

Answer: D

The cubes must fit a whole number of times along each edge, so their edge length must divide 6, 12 and 15. To make the fewest cubes, take the largest such edge - the highest common factor.

\[ \text{HCF}(6, 12, 15) = 3 \]

\[ n = \frac{6}{3} \times \frac{12}{3} \times \frac{15}{3} = 2 \times 4 \times 5 = 40 \]

Equivalently \( \tfrac{6 \times 12 \times 15}{3^{3}} = \tfrac{1080}{27} = 40 \). Fewest cubes means largest cubes - the two requirements pull in opposite directions.

Answer: B

The water leaving each minute forms a prism whose cross-section is the river's channel and whose length is the distance travelled in that minute.

\[ \text{cross-section} = 1.5 \times 36 = 54 \text{ m}^{2} \]

\[ \text{speed} = 3.5 \text{ km/h} = \frac{3500}{60} = 58.33 \text{ m/min} \]

\[ V = 54 \times 58.33 = 3150 \text{ m}^{3} \text{ per minute} \]

Converting km/h to metres per minute, not per second, is what matches the question's units; per hour the figure would be 189,000 m³.

Answer: C

Melting conserves volume, so divide the target cylinder's volume by one coin's. Halve both diameters first.

\[ V_{\text{coin}} = \pi (0.75)^{2}(0.2) = 0.1125\pi \text{ cm}^{3} \]

\[ V_{\text{cylinder}} = \pi (2.25)^{2}(10) = 50.625\pi \text{ cm}^{3} \]

\[ n = \frac{50.625\pi}{0.1125\pi} = \frac{50.625}{0.1125} = 450 \]

Using the diameters directly instead of the radii leaves the ratio unchanged only because both are squared - but the thickness is not, so it is safer to convert first.

Answer: B

A drinking tumbler is circular at both the rim and the base, but wider at the top than at the bottom, and neither end closes to a point.

That is a frustum of a cone - a cone with its tip sliced off by a plane parallel to the base.

A full cone would taper to a point, and a cylinder would have the same width top and bottom.

Answer: D

The cork nose of the shuttlecock is a rounded dome - a hemisphere. The feather skirt is circular at both ends, narrow where it meets the cork and wide at the open end, closing to a point at neither - a frustum of a cone.

A sphere would make the base fully round rather than flat against the feathers, and a plain cone would taper to a point instead of flaring open.

Answer: A

Cutting a cone with a plane parallel to its base produces two pieces: a smaller cone at the top, containing the apex, and the remaining piece below.

Removing the small cone leaves a solid with two parallel circular faces of different radii and a slanted curved surface joining them - a frustum of a cone.

The word comes from the Latin for "piece" or "morsel", which is exactly what the remaining part is.

Answer: A

A right circular cone has a circular cross-section at every level, shrinking steadily toward the apex.

A plane parallel to the base therefore cuts the cone in a circle - smaller than the base, but a circle nonetheless.

The frustum is the three-dimensional solid left below that cut, not the cross-section itself; the cross-section is a flat, two-dimensional figure.

Answer: A

Halve the diameter to get the radius, then apply the volume formula with \( \pi = 3.142 \).

\[ r = \frac{8}{2} = 4 \text{ cm}, \qquad h = 16 \text{ cm} \]

\[ V = \pi r^{2} h = 3.142 \times 16 \times 16 \]

\[ V = 3.142 \times 256 = 804.352 \text{ cm}^{3} \]

Using the diameter in place of the radius would quadruple the answer to about 3217 cm³.

Answer: B

Rearrange the volume formula for the radius, then double it.

\[ V = \pi r^{2} h \quad\Longrightarrow\quad r^{2} = \frac{V}{\pi h} \]

\[ r^{2} = \frac{900}{3.142 \times 20} = \frac{900}{62.84} = 14.32 \]

\[ r = \sqrt{14.32} = 3.785 \text{ cm} \]

\[ d = 2r = 7.57 \text{ cm} \]

The question asks for the diameter, so the square root must be doubled - stopping at the radius leaves 3.79 cm.