Question

Mathematics teacher of a school took her 10th standard students to show Red fort. It was a part of their Educational trip. The teacher had interest in history as well. She narrated the facts of Red fort to students. Then the teacher said in this monument one can find combination of solid figures. There are 2 pillars which are cylindrical in shape. Also 2 domes at the corners which are hemispherical.7 smaller domes at the centre. Flag hoisting ceremony on Independence Day takes place near these domes.

(i) How much cloth material will be required to cover 2 big domes each of radius 2.5 metres? (Take π = \(\frac{22}{7}\)) 

a) 75m² 

b) 78.57m² 

c) 87.47m² 

d) 25.8m²

(ii) Write the formula to find the volume of a cylindrical pillar. 

a) πr²h 

b) πrl 

c) πr(l + r) 

d) 2πr

(iii) Find the lateral surface area of two pillars if height of the pillar is 7m and radius of the base is 1.4m. 

a) 112.3cm² 

b) 123.2m² 

c) 90m² 

d) 345.2cm²

(iv) How much is the volume of a hemisphere if the radius of the base is 3.5m? 

a) 85.9 m³

b) 80 m³ 

c) 98 m³ 

d) 89.83 m³

(v) What is the ratio of sum of volumes of two hemispheres of radius 1cm each to the volume of a sphere of radius 2 cm? 

a) 1:1 

b) 1:8 

c) 8:1 

d) 1:16

Answer 1

(i) b) 78.57m² 

Cloth material required = 2 X S A of hemispherical dome

= 2 x 2πr² 

= 2 x 2 x \(\frac{22}{7}\) x (2.5)² m² 

= 78.57 m²

(ii) a) πr²h

Volume of a cylindrical pillar = πr²h

(iii) b) = 123.2 m²

Lateral surface area = 2 x 2πrh 

= 4 x \(\frac{22}{7}\) x 1.4 x 7 m² 

= 123.2 m²

(iv) d) 89.83 m³

Volume of hemisphere = \(\frac{2}{3}\)πr³ 

= \(\frac{2}{3}\frac{22}{7}\) (3.5)³ m³ 

= 89.83 m³

(v) b) 1:8

Sum of the volumes of two hemispheres of radius 1cm each = 2 x \(\frac{2}{3}\pi1^3\)

Volume of sphere of radius 2cm = \(\frac{4}{3}\pi2^3\)

So, required ratio is \(\frac{2\times\frac{2}{3}\pi1^3}{\frac{4}{3}\pi2^3}\)

= 1:8