Mensuration Questions and Answers

Answer: B

The sheet is 1 cm thick, so the outer radius is 5 cm and the inner 4 cm. The metal is the shell between them.

\[ V = \frac{2}{3}\pi\left(R^{3} - r^{3}\right) = \frac{2}{3}\pi(125 - 64) \]

\[ = \frac{2}{3}\pi(61) = \frac{122\pi}{3} \text{ cm}^{3} \]

\[ \text{weight} = \frac{122\pi}{3} \times 9 = 366\pi \text{ g} \]

Treating the hemisphere as solid would give \( \tfrac{2}{3}\pi(125) \times 9 = 750\pi \) g - the cavity must be subtracted.

Answer: D

Volume of the prism = 1056 cm³
WKT, Volume of prism = Base x Height
Base area = Area of trapezium
Area of trapezium = (1/2)(a + b)*h = (1/2)[8 + 14]*8
= (1/2)*22* 8 = 88 cm²

Volume of prism = (Area of trapezium) x Height
1056 = 88 * height
Height of prism = 12 cm.

Answer: A

Area of a circle = 4π cm²
πr² = 4π
r² = 4
r = 2 cm.
WKT, Radius of incircle = a/2√3
2 = a/2√3
a = 4√3.

WKT, Area of equilateral triangle = (√3/4)a²
Area of equilateral triangle = (√3/4)(4√3)²
= 12√3 cm².

Answer: D

A rectangular water tank is 80 m x 40 m. Water flows into it through a pipe of 40 sq.cm at the opening at a speed of 10 km/hr. The water level will rise in the tank in half an hour is

Volume of water filled by pipe in 30 minutes

-----> (40 *1000000/2) cu.cm
-----> = 2000000 cu.cm

Therefore, Height of water level

-----> (2000000/8000 *4000)
-----> = (5/8) cm.

Hence the answer is: (5/8) cm

Answer: B

Equal volumes with heights in the ratio \( 1 : 2 \):

\[ \pi r_{1}^{2} h = \pi r_{2}^{2} (2h) \]

\[ r_{1}^{2} = 2 r_{2}^{2} \quad\Longrightarrow\quad \frac{r_{1}}{r_{2}} = \sqrt{2} \]

So the ratio is \( \sqrt{2} : 1 \). The shorter cylinder must be wider, and because volume depends on \( r^{2} \), halving the height needs only a \( \sqrt{2} \) increase in radius rather than a doubling.

Answer: D

The longest bamboo lies along the room's space diagonal.

\[ d = \sqrt{l^{2} + b^{2} + h^{2}} = \sqrt{25 + 16 + 9} = \sqrt{50} \]

\[ = 5\sqrt{2} \approx 7.07 \text{ m} \]

The 7 m option is close but not exact; the floor diagonal alone would be \( \sqrt{41} = 6.4 \) m.

Answer: A

\[ A = \frac{\sqrt{3}}{4}a^{2} = \frac{\sqrt{3}}{4} \times 36 \]

\[ = 9\sqrt{3} \approx 15.6 \text{ sq. cm} \]

The side is squared before the constant is applied - multiplying 6 by \( \tfrac{\sqrt{3}}{4} \) directly would give \( 1.5\sqrt{3} \).

Answer: D

W.K.T, the shape of Kite resembles Quadrilateral.
The adjacent sides of a kite are of equal length.

Given, perimeter of Quadrilateral = 100 m
Length of one of the side = x = 20 m
Let the Length of other side = y

W.K.T, perimeter of Quadrilateral = sum of the length of all the four sides
---> 100 = 20 + 20 + y + y
---> 100 = 40 + 2y
---> 60 = 2y
---> y = 30 m

Hence, the length of sides of the kite (ie., Quadrilateral) = 20 m, 20 m, 30 m, 30 m

Answer: A

Given, ratio of the parallel sides of a trapezium = 5 : 3
---> Let the parallel sides (ie., 'a' and 'b') be 5x and 3x respectively.
Given, perpendicular distance = height = 24 m
and Area = 1440 m2

Formula: Area of Trapezium = (1/2) * h * (a + b)
---> 1440 = (1/2) * 24 * (5x + 3x)
---> 1440 = 12 * (8x)
---> 120 = 8x
---> 15 = x

Thus, parallel sides, a = 5x = 5 * 15 = 75 m
and b = 3x = 3 * 15 = 45 m
Hence, Longer parallel side is 'a' = 75 m
Smaller parallel side is 'b' = 45 m.

Answer: D

Let the side of square be 2a.
So, the diameter of circle = 2a
Radius of circle = a

WKT, Area of square = 4a
Area of circle = π r²
Perimeter of circle = 2π r
Area of rectangle = l x b
Perimeter of rectangle = 2(l + b)
Given that difference between area of the square and circle is 262.5 cm².
Area of square - Area of circle = 262.5
4a² - (22/7)a² = 262.5
28a² - 22a² = 1837.5
6a² = 1837.5
a² = 306.25
a = 17.5 cm

Perimeter of circle = 2πa = 2 x (22/7) x 17.5 = 110 cm

Also given length of rectangle is 20% more than the breadth of rectangle.
Let the breadth of rectangle be X.
So, length of rectangle = (120/100)X = 1.2X
As per the question,
Perimeter of rectangle = Perimeter of circle
2(1.2X + X) = 110
2.4X + 2X = 110
4.4X = 110
X = 25 cm
Breadth of rectangle = 25 cm
Length of rectangle = 1.2(25) = 30 cm

Area of rectangle = 25 x 30 = 750 cm².