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Question Number 178884 by mr W last updated on 22/Oct/22

Commented by mr W last updated on 22/Oct/22

a goat is tied at a point on the foot of  a hill which has the shape of a sphere   with radius 60m and  height 20m as   shown. the length of the rope is 20m.  find the total area where the goat   can graze.

agoatistiedatapointonthefootofahillwhichhastheshapeofaspherewithradius60mandheight20masshown.thelengthoftheropeis20m.findthetotalareawherethegoatcangraze.

Commented by mr W last updated on 23/Oct/22

give a try sir?

giveatrysir?

Commented by Rasheed.Sindhi last updated on 23/Oct/22

Sorry sir!

Sorrysir!

Answered by mr W last updated on 23/Oct/22

Commented by mr W last updated on 23/Oct/22

a part of the area where the goat can  graze in on the ground (A_1 ).  the other part is on the hill (sphere  surface) (A_2 ).

apartoftheareawherethegoatcangrazeinontheground(A1).theotherpartisonthehill(spheresurface)(A2).

Commented by mr W last updated on 23/Oct/22

Commented by aleks041103 last updated on 23/Oct/22

A_1  is comprised of 2 shapes.  1 is half a circle  and  2 is some strange shape the rope traces out  when the touching point between the rope  and the hill moves.

A1iscomprisedof2shapes.1ishalfacircleand2issomestrangeshapetheropetracesoutwhenthetouchingpointbetweentheropeandthehillmoves.

Commented by mr W last updated on 23/Oct/22

cos α=((20)/(2×60))=(1/6)  β=π−2α  sin β=sin 2α=2×(1/6)×((√(35))/6)=((√(35))/(18))  A_1 =(1/2)(2π−2α)×20^2 −2×((60^2 )/2)×(β−sin β)  A_1 =(π−α)×20^2 −60^2 ×(π−2α−((√(35))/(18)))     =(π−cos^(−1) (1/6))×20^2 −60^2 ×(π−2 cos^(−1) (1/6)−((√(35))/(18)))     =(2×60^2 −20^2 ) cos^(−1) (1/6)−(60^2 −20^2 )π+60^2  ((√(35))/(18))     ≈672.887 m^2

cosα=202×60=16β=π2αsinβ=sin2α=2×16×356=3518A1=12(2π2α)×2022×6022×(βsinβ)A1=(πα)×202602×(π2α3518)=(πcos116)×202602×(π2cos1163518)=(2×602202)cos116(602202)π+6023518672.887m2

Commented by mr W last updated on 23/Oct/22

Commented by mr W last updated on 23/Oct/22

(R−20)^2 +60^2 =R^2   ⇒R=((20^2 +60^2 )/(40))=100 m  AC^(⌢) =AB^(⌢) =AD^(⌢) =20 m=rope length  .......

(R20)2+602=R2R=202+60240=100mAC=AB=AD=20m=ropelength.......

Commented by Tawa11 last updated on 23/Oct/22

Great sir

Greatsir

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