Question

Mathematics teacher of a school took her 10th standard students to show Tajmahal.

The teacher said in this monument one can find combination of solid figures.

(i)Write the formula to find the volume of a hemispherical dome.

(ii)Find the lateral surface area of 4 pillars if height of each pillar is 20m and radius of the

base is 1.4m.

(iii)How much is the volume of 2 small hemispheres each of radius 2.1m?

(iv)How much cloth material will be required to cover 1 big dome of radius 4.2m?

(v)What is the ratio of sum of volumes of four hemispheres of radius 1m each to the

volume of a sphere of radius 2m?​

Answer 1

Given:

The teacher said in Taj Mahal one can find a combination of solid figures.

To find:

Write the formula to find the volume of a hemispherical dome.

Find the lateral surface area of 4 pillars if the height of each pillar is 20m and a radius of the base is 1.4m.

How much is the volume of 2 small hemispheres each of radius 2.1m?

How much cloth material will be required to cover 1 big dome of radius 4.2m?

What is the ratio of the sum of volumes of four hemispheres of radius 1m each to the volume of a sphere of radius 2m?​

Solution:

Finding the formula of calculating the volume of the hemispherical dome:

The volume of the hemispherical dome is,

= Volume of a hemisphere

= $\boxed{\bold{\frac{2}{3} \pi r^3}}$

Finding the lateral surface area of 4 pillars:

The height of the cylindrical pillar = 20 m

The radius of the base of the cylindrical pillar = 1.4 m

∴ The lateral surface area of 4 cylindrical pillars are,

= 4 × Lateral surface area of each cylindrical pillar

= 4 × $2 \pi r h$

= $4 \times 2 \times \frac{22}{7} \times 1.4 \times 20$

= $4 \times 2 \times 22 \times 0.7 \times 20$

= $\boxed{\bold{704\:m^2}}$

Finding the volume of 2 small hemispheres:

The radius of each hemisphere = 2.1 m

∴ The volume of 2 small hemispheres are,

= 2 × Volume of each hemisphere

= 2 × $\frac{2}{3} \pi r^3$

= $2 \times \frac{2}{3} \times \frac{22}{7} \times 2.1^3$

= $\boxed{\bold{38.8\:m^3}}$

Finding the required area of the cloth to cover the dome:

The radius of the big dome = 4.2 m

∴ The area of cloth required to cover the big dome is,

= Curved surface area of the hemisphere

= $2 \pi r^2$

= $2 \times \frac{22}{7} \times 4.2 ^2$

= $\boxed{\bold{110.88\: m^2 }}$

Finding the ratio of the volume of 4 hemispheres to a sphere:

The radius of each hemisphere = 1 m

The radius of sphere = 2 m

∴ The ratio of the volume of 4 hemispheres to a sphere is,

= $\frac{Volume \:of\: 4\: hemisphere}{Volume\: of \: the \:sphere}$

= $\frac{4\times \frac{2}{3}\pi r^3 }{\frac{4}{3} \pi r^3}$

= $\frac{4\times 2\times 1^3 }{4 \times 2^3}$

= $\frac{ 2}{2^3}$

= $\frac{1}{4}$

= $\boxed{\bold{1 : 4 }}$

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