Q) A golf ball is spherical with about 300 – 500 dimples that help increase its velocity while in play. Golf balls are traditionally white but available in colours also. In the given figure, a golf ball has diameter 4.2 cm and the surface has 315 dimples (hemi-spherical) of radius 2 mm.

Based on the above, answer the following questions :
(i) Find the surface area of one such dimple.
(ii) Find the volume of the material dug out to make one dimple.
(iii) (a) Find the total surface area exposed to the surroundings.
OR
(iii) (b) Find the volume of the golf ball.

Ans: 

Given that the diameter of the ball = 4.2 cm = 42 mm

Therefore, the radius of the ball = 21 mm

radius of the dimple = 2 mm

(i) Surface area of one dimple:

We can see that the dimple is hemispherical and the surface area of a hemisphere = 2 r²

= = 25.1 mm²

Therefore, the surface area of one dimple is 25.1 mm².

(ii) Volume of material dug out of one dimple:

The material dug out to make one dimple = volume of hemisphere

We know that the volume of a hemisphere = r³

= = 16.76 mm³

Therefore, the volume of the material dug out to make one dimple is 16.76 mm³.

(iii) (a) Total surface area exposed to the surroundings:

The surface area exposed to the surroundings = Curved surface area of ball – Base area cut by 315 dimples + Curved surface area added by 315 dimples

= 4 (21)²  –  315 (2)²  + (315) 2 (2)² 

= 1764  – 1260  + 2520

= 3024 = (3024) = 9504 mm²

Therefore, the total surface area exposed to the surroundings is 9504 mm².

(iii) (b) The volume of the golf ball:

The volume of a golf ball = Volume of ball sphere – volume of 315 hemispherical dimples

= – 315

= 12348 – 1680 = 10668

= (10668) = 33528 mm³ 

Therefore, the volume of the golf ball is 33,528 mm³.