规则圆柱体,直径为d,水桶长为H,横着放,往里面灌水,水面高为h,求桶里水的体积写上公式吧!谢谢了具体一点,解释的详细一点.

Question

规则圆柱体,直径为d, 水桶长为H,横着放,往里面灌水,水面高为h,求桶里水的体积
写上公式吧!谢谢了 具体一点,解释的详细一点.

Answer 1

①当h≤d/2时,假设水面与圆面交点为A,B,圆心为O, 又设小的扇形ABO的圆心角为θ,则 V水=V小扇ABO-V柱ABO =V小扇ABO-S△ABO*H =θ/2π *S⊙O* H - S△ABO*H =(θ/2π *S⊙O - S△ABO)*H ∵θ=2arcsin(d/2-h)/(d/2)=2arcsin(d-2h)/d S△ABO=1/2*(d/2)θ=(d/2-h)√[(d-h)h] ∴V水={[(2arcsin(d-2h)/d) /2π *π(d/2)^2 ] - (d/2-h)√[(d-h)h]}H ={d^2 arcsin(d-2h)/d) - (d/2-h)√[(d-h)h]}H ②当h>d/2时,假设水面与圆面交点为A,B,圆心为O, 又设小的扇形ABO的圆心角为θ,则大的扇形ABO的圆心角为2π-θ ∴V水=V大扇ABO+V柱ABO =V大扇ABO+S△ABO*H =(2π-θ)/2π *S⊙O* H + S△ABO*H =(2π-θ/2π *S⊙O + S△ABO)*H ={[(2π-2arcsin(d-2h)/d) /2π *π(d/2)^2 ] + (d/2-h)√[(d-h)h]}H ={[2π-arcsin(d-2h)/d)]d^2 + (d/2-h)√[(d-h)h]}H
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