Sphere and Hemisphere-RRB

Sphere and Hemisphere-RRB Sphere and Hemisphere-RRB

SPHERE

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In the following formulae, r = radius of sphere, d = diameter of sphere = 2r
• Surface area of a sphere = 4πr² = πd²

• Volume of a sphere = (4/3) πr³= (1/6)πd³

Example – 1

Calculate the volume of sphere with radius 4 cm.

Answer

Volume of sphere = (4/3) πr³

= (4/3) π 4³

= (4/3) $\times$ (22/7) $\times$ 4 $\times$ 4 $\times$ 4

= 268.19 cm²

Example – 2

The volume of a spherical ball is 5,000 cm³. What is the radius of the ball?

Answer

According to question V = (4/3) πr³= 5000 cm³

=> r³= 5000 x 3/4 $\times$ 7/22

=> r³=1193.18 cm³= (10.61)³

Hence r = 10.61

HEMISPHERE

In the following formulae, r = radius of sphere
• Volume of a hemisphere =(2/3)πr³
• Curved surface area of a hemisphere = 2πr²
• Total surface area of a hemisphere = 3πr²

Example – 3

The radius of hemispherical balloon increase from 7 cm to 14 cm as air is being pumped into it. Find the ratios of the surface areas of the balloon in two cases.

Answer

For 1st hemisphere, r = 7 cm
Total Surface Area = 3 π r² = 3   $\times$ π   $\times$ 7²
Total Surface Area of hemisphere= 3 π $\times$ 7²
Total Surface Area after increase = 3π   $\times$ 14²
⇒ S 1 : S 2 = 3 π   $\times$ 7²: 3π   $\times$ 14² = 1 : 4

Example – 4

Show that the surface area of sphere is same as that of the lateral surface area of a cylinder that just encloses the sphere.

Answer

Total surface area of sphere = 4 π r² ——(1)
The radius and height of the cylinder that just encloses the sphere of radius r and 2r respectively.
∴ Curved Surface Area of cylinder = 2 π r h
= 2 π r $\times$ 2r (Since h = 2r)
∴ Curved Surface Area of cylinder = 4 π r²—-(2)
∴ From (1) and (2)
Surface area of sphere is same as that of the lateral surface area of a cylinder that just encloses the sphere.

Question 5: A sphere and a hemisphere have the same surface area. The ratio of their volumes is

\frac{\sqrt{3}}{4} : 1

3 \frac{\sqrt{3}}{4} : 1

\frac{\sqrt{3}}{8} : 1

D) 3√3/8 : 1

Solution: Answer is D

Let r and R are the radius of two spheres

Given surface area of sphere = Surface area of hemisphere

So 4πr² = 3πR²

$\Rightarrow$ r² / R² = $\frac{3}{4}$

$\Rightarrow$ $\frac{r}{R}$ = √3/2

Hence the ratio of volumes is (4/3) πr³/ (4/3) πR³

= r³/ R³

=( $\frac{r}{R}$ )³ = 3√3/8

Hence ratio is 3√3/8 : 1