Volume & Surface Areas Questions and Answers

Answer: B

The surface area is 784 pi square centimetres. The material in one shell is \[\frac{4}{3}\pi(5^3 - 3^3)\] \[= \frac{4}{3}\pi \times 98\] For 28 shells it is \[\frac{4}{3}\pi \times 98 \times 28\] \[= \frac{4}{3}\pi \times 2744\] The new solid sphere therefore has \(r^3 = 2744\), so r = 14 cm. Its surface area is \[4\pi r^2 = 4\pi \times 196 = 784\pi\] The other options correspond to radii of 12, 13 and 15 cm, which do not give 2744 as r cubed (12 cubed is 1728, 13 cubed is 2197 and 15 cubed is 3375).

Answer: C

The capacity is 2.42 pi litres. The base perimeter is \(2\pi r = 22\pi\), so r = 11 cm. The curved surface area is \(\pi r l = 671\pi\), so \(l = 671/11 = 61\) cm. The height is \[h = \sqrt{61^2 - 11^2} = \sqrt{3600} = 60\] cm. The volume is \[\frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \times 121 \times 60\] \[= 2420\pi\ \text{cm}^3\] Since 1 litre is 1000 cubic centimetres, this is 2.42 pi litres. The other options come from a wrong height or from forgetting the conversion to litres.

Answer: C

The cost is Rs. 2,385. The box has no lid, so the outer surface to colour is the base plus the four sides: \[lb + 2h(l + b) = 90 \times 70 + 2 \times 30 \times 160\] \[= 6300 + 9600 = 15900\ \text{cm}^2\] At 15 paise per square centimetre the cost is 15900 x 0.15 = Rs. 2,385. If the lid were also coloured, the area would be 22,200 cm squared and the cost Rs. 3,330, which is not an option.

Answer: A

The base diameter is 15 cm, so the radius is 7.5 cm and the slant height is 12.5 cm. The height follows from the right triangle.

\[ h = \sqrt{12.5^2 - 7.5^2} = \sqrt{100} = 10 \]

\[ V = \tfrac{1}{3}\pi (7.5)^2 (10) = 187.5\pi \]

The volume is 187.5 pi cubic centimetres. The option 562.5 pi is three times as much and comes from leaving out the factor one third.

Answer: B

The first cone has radius 12 cm and slant height 37 cm, so its height is the square root of 37 squared minus 12 squared, which is 35 cm.

\[ \tfrac{1}{3}\pi (12)^2 (35) = \tfrac{1}{3}\pi (6)^2 h \]

\[ 144 \times 35 = 36h \Rightarrow h = 140 \]

The new cone is 140 cm high. The height of 148 cm would come from taking the volume in terms of the slant height, and 150 and 160 cm do not fit.

Answer: A

8000 small spheres are formed. Melting and recasting keeps the total volume the same, so the number of small spheres equals the ratio of the volumes. Since the volume of a sphere is proportional to the cube of its radius, the common factor \(\frac{4}{3}\pi\) cancels.

The radius ratio is 20, so the count is 20 cubed. Using 20 squared would give 400, and none of 800, 1600 or 16000 is 20 cubed.

Answer: B

The height is 4.75 m. Volume = length × width × height, so \[ h = \frac{380}{10 \times 8} = 4.75 \text{ m} \]

3.75, 5.75 and 6.75 m would hold 300, 460 and 540 cubic metres, not 380.

In general, volume = length x width x height, so dividing the volume by the area of the base (10 x 8 = 80 square metres) gives the height directly.

Answer: C

The wire is 18 m long. The volume of the sphere equals the volume of the wire. Radius of the sphere 6 cm, radius of the wire 0.4 cm: \[ \frac{4}{3}\pi(6)^{3} = \pi(0.4)^{2}h \] \[ 288 = 0.16h \Rightarrow h = 1800 \text{ cm} = 18 \text{ m} \]

The other options come from using the wire diameter as the radius or slipping a unit.

Answer: B

The ratio of heights is 8 : 3. The volume of a cone is one third of pi r squared h, so h is proportional to V divided by r squared. With V in the ratio 2 : 3 and r in the ratio 1 : 2, h1 : h2 = (2/1) : (3/4) = 2 : 0.75 = 8 : 3. The ratio 2 : 1 ignores the radius, 3 : 8 is inverted and 3 : 2 uses the volumes only.

Answer: A

The ratio of surface areas is 9 : 1. The volume of a sphere is proportional to the cube of its radius, so a volume ratio of 27 : 1 gives a radius ratio of 3 : 1. The surface area is proportional to the square of the radius, so the areas are in the ratio 9 : 1. 1 : 9 is inverted, 6 : 1 and 3 : 1 do not follow from squaring the radius ratio. The ratio of the radii is the cube root of 27 over the cube root of 1, which is 3 : 1. Surface area depends on the square of the radius, so the areas are in the ratio 9 : 1; the option 3 : 1 is the radii ratio, not the areas.