Mensuration Questions and Answers

Answer: D

The immersed cube displaces its own volume of water, which spreads over the vessel's base.

\[ V_{\text{cube}} = 15^{3} = 3375 \text{ cm}^{3} \]

\[ \text{base area} = 20 \times 15 = 300 \text{ cm}^{2} \]

\[ \text{rise} = \frac{3375}{300} = 11.25 \text{ cm} \]

The cube's own 15 cm edge is not the rise: the water spreads over a base twice the cube's footprint, so it climbs less far.

Answer: A

An increase of \( 9.09\% \) is a factor of \( \tfrac{1}{11} \), since \( \tfrac{1}{11} = 0.0909 \). The radius therefore becomes \( \tfrac{12}{11}r \).

Surface area varies as the square of the radius:

\[ \frac{S'}{S} = \left(\frac{12}{11}\right)^{2} = \frac{144}{121} = 1.1901 \]

\[ \text{increase} = 19.01\% \approx 19\% \]

Recognising 9.09% as \( \tfrac{1}{11} \) keeps the arithmetic exact; the decimal route gives 18.998% and invites a rounding error.

Answer: A

Correct answer: 240 sq. cm

\[\text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}\]

\[= \frac{1}{2} \times (15 + 25) \times 12 = \frac{1}{2} \times 40 \times 12 = 240 \text{ sq. cm}\]

The height must be the perpendicular distance between the parallel sides, not the length of a slanting side.

Answer: B

Correct answer: 154 sq. cm

Find the radius from the circumference first.

\[2\pi r = 44 \quad\Longrightarrow\quad r = \frac{44 \times 7}{2 \times 22} = 7 \text{ cm}\]

\[\text{Area} = \pi r^2 = \frac{22}{7} \times 49 = 154 \text{ sq. cm}\]

Answer: B

The rain forms a thin slab over the ground.

\[ 5 \text{ cm} = 0.05 \text{ m}, \qquad 1.5 \text{ hectares} = 15{,}000 \text{ m}^{2} \]

\[ V = 15{,}000 \times 0.05 = 750 \text{ m}^{3} \]

A hectare is 10,000 m², and the depth must be in metres to match.

Answer: D

In \( C = 2\pi r \), the letter \( r \) is the radius - the distance from the centre to any point on the circle.

The expression \( 2\pi r \) is itself the circumference, so \( r \) cannot be that; area and perimeter are likewise the results of formulas, not the input to this one.

Answer: B

The circumference of the circle is equal to the permeter of the rectangle. 

Let l = 6x and b = 5x 2(6x + 5x) = 2 * 22/7 * 3.5 

 =>  x = 1 

Therefore l = 6 cm and b = 5 cm Area of the rectangle = 6 * 5 = 30 cm²

Answer: C

Correct answer: 616 sq. cm

\[\text{Surface area} = 4\pi r^2 = 4 \times \frac{22}{7} \times 49 = 4 \times 154 = 616 \text{ sq. cm}\]

Notice that this is exactly four times the area of a great circle of the sphere, which is a useful check.

Answer: D

Cylinder: Radius - 24.5 cm; Height - 5 cm
Cuboid: Length - 7 cm; Breadth - 8 cm
WKT, Volume of cylinder = πr²h
Volume of cubiod = l x b x h

Volume of 80% of Cognac in cylindrical vessel = Volume of 9 cuboidal vessels
(22/7) x 24.5 x 24.5 x 5 x 0.8 = 9 x 7 x 8 x h
h = [(22/7) x 24.5 x 24.5 x 5 x 0.8 ]/[9 x 7 x 8]
h = 7546/504
h = 14.9 cm.

Answer: B

Find the radius from the circumference, then use it for the area.

\[ 2\pi r = 64\pi \quad\Longrightarrow\quad r = 32 \text{ cm} \]

\[ A = \pi r^{2} = \pi (32)^{2} = 1024\pi \text{ cm}^{2} \]

The \( \pi \) cancels in the first step and returns in the second, so no numerical value for it is ever needed.