PLZ ANYONE HELP ME IN SOLVING THIS QUESTION ........  A GOLF BALL HAS DIAMETER EQUAL TO 4.1CM. ITS SURFACE HAS 150 DIMPLES EACH OF RADIUS 2MM (0.2 CM) .CALCULATE TOTAL SURFACE AREA WHICH IS...

PLZ ANYONE HELP ME IN SOLVING THIS QUESTION ........

 

A GOLF BALL HAS DIAMETER EQUAL TO 4.1CM. ITS SURFACE HAS 150 DIMPLES EACH OF RADIUS 2MM (0.2 CM) .CALCULATE TOTAL SURFACE AREA WHICH IS EXPOSED TO SURROUNDINGS ASSUMING THAT THE DIMPLES ARE HEMISPHERCAL.

 

THE ANSWER IS 71.68 CM^2 (CENTIMETRE SQUARE). BUT I DON'T HOW TO  WANT STEP BY STEP .......

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beckden | High School Teacher | (Level 1) Educator

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The surface area of a sphere is `4pir^2` . The diameter is 4.1 so the radius is 2.05.  so if the golf ball was without dimples it would have a surface area of `4pi(2.05)^2=16.81pi` cm^2

Now a dimple has a surface area of `1/2(4pi(0.2)^2)=0.08pi` cm^2

But we cannot just add 150 dimples, they replace some of the sphere's surface area so we have to add 150 dimples and subtract 150 circles of radius 0.2 cm on the sphere.

The surface area of a spherical cap of radius a on a sphere of radius r is `A=2pir(r-sqrt(r^2-a^2))` , in our case r=4.1cm and a=0.2cm. So each spherical cap (where the dimple will go) has an area of `2pi(2.05)(2.05-sqrt(2.05^2-0.2^2))=0.040095637pi`

So `16.81pi+150(0.08-0.040095637)pi=22.795654pi=71.614cm^2`

If you consider do not consider the exact value for the spherical cap and approximate it as the area of a circle, `2pia=0.04pi` , you get

`16.81pi+150(0.08-0.04)pi=22.81pi=71.659728cm^2`

The answer you gave seems a little high.

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